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Problem Solving
Caution, This Induction May Induce Vomiting
2 3 4
3 4 5
1. Observe that 1 2 2 3
, 1 2 2 3 3 4
, and
3
3
456
.
1 2 2 3 3 4 4 5
3
Use inductive reasoning to make a conjecture about the value
1 2 2 3 3 4 n n 1 .
Use your conjecture to determine the value of 1 2 2 3 3 4 100,000 100,001.
of
Oh Brother! No, Oh Sister!
2. A boy has twice as many sisters as brothers, and each sister has two more sisters than
brothers. How many brothers and sisters are in the family?
{Hint: Let b be the number of boys and g the number of girls. Now write down some
equations.}
Exactly How Do You Want Your Million?
3. Find a positive number that you can add to 1,000,000 that will give you a larger value than if
you multiplied this number by 1,000,000? Find all such numbers.
{Hint: Let the positive number be x, and solve x 1,000,000 1,000,000x .}
Interesting Is In The Eye Of The Beholder
4. There is an interesting five-digit number. With a 1 after it, it is three times as large as with a
1 before it. What is the number?
{Hint: If x is the five-digit number, then x abcde, abcde1 10 x 1,1abcde 100,000 x .}
Twenty-one, But Not Blackjack
5. Find the 21-digit number so that when you write the digit 1 in front and behind, the new
number is 99 times the original number.
{Hint: If x is the 21-digit number then x abcdefghijklmnopqrstu ,
and abcdefghijklmnopqrstu1 10 x 1,
and 1abcdefghijklmnopqrstuv1 10,000,000,000,000,000,000,000 10 x 1 }
Stand On Your Heads And Get It Together
6. The sum of two numbers is 50, and their product is 25. Find the sum of their reciprocals.
{Hint: x y 50 , xy 25 , so divide the first equation by the second equation.}
The Last Two Standing
7. What are the final two digits of 71997 ?
{Hint: Look for a pattern:
Power of 7
7 2 49
73 343
74 2401
75 16807
76 117649
Final two digits
49
43
01
07
49
}
Don’t Give Up; Don’t Ever Give Up!
f x 1
8. Given that f 11 11 and f x 3
for all x , find f 2000 . First find
f x 1
f 14 , f 17 , f 20 ,
.
{Hint: Look for a pattern:
n
11
f(n)
11
14
5
6
1
11
17
20
23
6
5
11
}
Mind Your Four’s And Two’s
9. What is the value of x if 4 4200 2x ?
200
{Hint: Factor 4200 4200 and use the fact that 4 22 .}
A Lot Of Weeks, But How Many Days Left Over?
10. What is the remainder when 215,110 is divided by 7?
{Hint: Look for a pattern in the remainders:
Power of 2 Remainder when divided by 7
2
21 2
4
22 4
3
1
2 8
2
24 16
5
4
2 32
}
Can You Just Tell Me How Old Your Children Are!
11. A student asked his math teacher, “How many children do you have, and how old are they?”
“I have 3 girls,” replied the teacher. “The product of their ages is 72, and the sum of their
ages is the same as the room number of this classroom.” Knowing that number, the student
did some calculations and said, “There are two solutions.” “Yes, that is so,” said the
teacher, “but I still hope that the oldest will some day win a math prize at this school.” The
student then gave the ages of the three girls. What are the ages?
{Hint:
Triple factors of 72
1,1,72
1,2,36
1,3,24
1,4,18
1,6,12
1,8,9
2,2,18
2,3,12
2,4,9
2,6,6
3,3,8
3,4,6
Sum of the factors
74
39
28
23
19
18
22
17
15
14
14
13
}
Cover All Your Bases, If It’s Within Your Power.
12. Solve for x if x 5 x 5
2
x 2 9 x 20
1.
{Hint: Any number raised to the zero power, except zero itself, equals 1. 1 raised to any
power is equal to 1. -1 raised to an even power is equal to 1.}
Who Needs Logarithms?
13. If 2 15 and 15 32 , then find the value of xy .
x
y
{Hint: Substitute the first equation into the second equation, and use an exponent property.}
I Refuse To Join Any Club That Would Have Me As A Member.
14. A club found that it could achieve a membership ratio of 2 Aggies for each Longhorn either
by inducting 24 Aggies or by expelling x Longhorns. Find x.
{Hint: Let L be the number of Longhorns and A be the number of Aggies, to get
2 A 24 2
A
.}
,
1
L
1 Lx
I Cannot Tell A Fib(onacci), My Name Is Lucas.
15. If f 1 1 , f 2 2 , and f n f n 1 f n 2 for n 3,4,5,
a) What is the value of f 12 ?
{Hint: f 3 f 2 f 1 1 2 3 , f 4 f 3 f 2 3 2 5 ,…Keep going.}
Amazingly, f can be represented as f n x n y n for n 1,2,3,4,5,
b) Find the values of x and y .
{Hint: f 1 x y, f 1 1 , f 2 x 2 y 2 , f 2 2 , this should be enough to find values
of x and y.}
Seven Heaven or Seven…
16. Find the largest power of 7 that divides 343!. 343! 1 2 3 4
342 343
{Hint: The multiples of 7 occurring in the expansion of 343! are 7,14,21,28, ,7 49 .
The multiples of 7 2 49 occurring in the expansion of 343! are 49,98, ,49 7
The multiple of 73 343 occurring in the expansion of 343! is just 343.
There are no multiples of higher powers of 7 occurring in the expansion of 343!}
A Special Case Of The Chinese Remainder Theorem
17. The positive integer n, when divided by 3, 4, 5, 6, and 7, leaves remainders of 2, 3, 4, 5, and
6, respectively. Find the smallest possible value of n.
{Hint: n 3a 2, n 4b 3, n 5c 4, n 6d 5, n 7e 6 .
This means that
n 1 3 a 1 , n 1 4 b 1 , n 1 5 c 1 , n 1 6 d 1 , n 1 7 e 1 .
So
n 1 is a common multiple of 3, 4, 5, 6, and 7. What’s the least common multiple?}
How Low Can It Go?
18. The grades on six tests all range from 0 to 100 inclusive. If the average for the six tests is
93, what is the lowest possible grade on any one of the tests?
T1 T2 T3 T4 T5 T6
T1 T2 T3 T4 T5 T6 558
6
T1 558 T2 T3 T4 T5 T6 , so T1 will be as small as possible when
T2 T3 T4 T5 T6 is as large as possible.}
{Hint: 93
Just Your Average Joe.
19. If Joe gets 97 on his next math test, his average will be 90. If he gets 73, his average will be
87. How many tests has Joe already taken?
{Hint: Let n be the number of tests he has already taken, and T the total number of points he
T 97
T 73
has already earned on the tests. Then
90,
87 .}
n 1
n 1
A Whole Lotta Zeroes
20. How many zeroes are at the end of the number 127! ? 127! 1 2 3 4
126 127
{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s. See the hint
for problem #16.}
2421
21. Find the ones digit of 13
{Hint: Look for a pattern:
Powers of 13
131
132
133
134
135
136
The Last One Standing
173783 .
One’s-digit
3
9
7
1
3
9
Powers of 17
171
17 2
173
17 4
175
176
One’s digit
7
9
3
1
7
9
}
Happy 2009!
22. Find the 2009th digit in the decimal representation of
{Hint:
1
.
7
1
0.142857 , so use a pattern.}
7
Zero, The Something That Stands For Nothing.
23. How many zeroes are at the end of the number 2300 5600 4400 ?
{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s.}
Looky Here Son, This Is A Problem, Not A Chicken.
24. Foghorn C sounds every 34 seconds, and foghorn D sounds every 38 seconds. If they sound
together at noon, what time will it be when they next sound together?
Foghorn C
Foghorn D
12:00
12:00
sound
together
12:00:34
12:01:08
12:00:38
12:01:42
12:01:16
{Hint: Every time they sound together after noon will have to be both a multiple of 34
seconds after noon and a multiple of 38 seconds after noon.}
A European Sampler
25. A box contains 8 French books, 12 Spanish books, 9 German books, 15 Portuguese books,
and 18 Italian books. What is the fewest number of books you can select from the box
without looking to be guaranteed of selecting at least 10 books of the same language?
{Hint: What is the largest number of books you can select and still not have 10 books of the
same language? The answer to the problem is 1 more than the answer to the
previous question.}
I Hope That You Are A Digital Computer?
26. What is the ones digit of 1! 2! 3! 999!?
{Hint: See what happens to the one’s digits:
Factorial One’s-digit One’s digit of the sum
1!
1
1
2!
2
3
3!
6
9
4!
4
3
}
27. Find the exact value of
The Collapse Of Rationalism
1
1
1
1
.
1 2
2 3
3 4
999,999 1,000,000
{Hint: Rationalize the denominators. For example:
1
1 2
1 2
2 1 .}
1
1 2 1 2
The Beast With Many Fingers And Toes
28. How many digits does the number 86,666 520,000 have?
{Hint: Zeroes come from factors of 10. Factors of 10 come from 5’s and 2’s.}
Don’t Get Stumped; Use The Fundamental Theorem Of Arithmetic.
29. Forrest stump heard that there are only two numbers between 2 and 300,000,000,000,000
which are perfect squares, perfect cubes, and perfect fifth powers. He decided to look for
them, and so far he has checked out every number up to about 100,000 and is beginning to
get discouraged. What are the numbers he is trying to find?
{Hint: Every positive whole number greater than 1 can be written as a product of prime
factors.
If N is a positive whole number greater than 2, then
n1
n2
N 2 3 5n3 pk nk . In order for N to be a perfect square, all the positive
exponents would have to be multiples of 2; in order for N to be a perfect cube, all
the positive exponents would have to be multiples of 3; and in order for N to be a
perfect fifth power, all the positive exponents would have to be multiples of 5. So all
the positive exponents would have to be common multiples of 2, 3, and 5.}
Sorry, I Can’t Give You Change For A Dollar.
30. What is the largest amount of money in U.S. coins(pennies, nickels, dimes, quarters, but no
half-dollars or dollars) you can have and still not have change for a dollar?
{Hint: It’s more than 99 cents. For instance: 3 quarters and 3 dimes is $1.05, but you can’t
make change for a dollar.}
Destination Cancellation.
31. Express as a fraction, in lowest terms, the value of the following product of 1,999,999
1
1 1 1
factors 1 1 1 1
.
2 3 4
2,000,000
{Hint: Look for a pattern:
1
1
2
1 1 1 2
1 1
2 3 2 3
1 1 1 1 2 3
1 1 1
2 3 4 2 3 4
1
2
1
3
1
4
}
Officer, I Got The License Plate Number, But I Was Lying On My Back.
32. The number on a license plate consists of five digits. When the license plate is turned
upside-down, you can still read a number, but the upside-down number is 78,633 greater
than the original license number. What is the original license number?
{Hint: The digits that make sense when viewed upside-down are 0, 1, 6, 8, and 9.
1st digit
2nd digit
3rd digit
4th digit
5th digit
7
8
6
3
3
Upside-down plate
Original plate
Difference of the plates
}
One Smokin’ Good Problem
33. Mrs. Puffem, a heavy smoker for many years finally decided to stop smoking altogether.
“I’ll finish the 27 cigarettes I have left,” she said to herself, “and never smoke another one.”
It was Mrs. Puffem’s practice to smoke exactly two-thirds of each complete cigarette(the
cigarettes are filterless). It did not take her long to discover that with the aid of some tape,
she could stick three butts together to make a new complete cigarette. With 27 cigarettes on
hand, how many complete cigarettes can she smoke before she gives up smoking forever,
and what portion of a cigarette will remain?
{Hint: With 27 complete cigarettes, she can smoke 27 complete ones and assemble 9 new
complete ones…, keep going.}
Just gimme an A.
34. A class of fewer than 45 students took a test. The results were mixed. One-third of the
class received a B, one-fourth received a C, one-sixth received a D, one-eight of the class
received an F, and the rest of the class received an A. How many students in the class got
an A?
{Hint: The number of students in the class must be a multiple of 3, 4, 6, and 8, and must be
smaller than 45.}
Working Backwards In Notsuoh
35. A castle in the far away land of Notsuoh was surrounded by four moats. One day the castle
was attacked and captured by a fierce tribe from the north. Guards were stationed at each
bride over the moats. Johann, from the castle, was allowed to take a number of bags of gold
as he went into exile. However, the guard at the first bridge took half of the bags of gold
plus one more bag. The guards at the second third and fourth bridges made identical
demands, all of which Johann met. When Johann finally crossed all the bridges, he had just
one bag of gold left. With how many bags of gold did Johann start?
{Hint: Sometimes working backwards is a good idea. If Johann has 1 bag of gold left, then
how many did he have when he approached the fourth guard?
x
12 x 1 1 }
approached the 4th guard
taken by the 4th guard
bags left
Grazin’ In The Grass Is A Gas, Baby, Can You Dig It?
36. A horse is tethered by a rope to a corner on the outside of a square corral that is 10 feet on
each side. The horse can graze at a distance of 18 feet from the corner of the corral where
the rope is tied. What is the total grazing area for the horse?
{Hint:
18
18
8
8
}
Life Is Like A Box Of Chocolate Covered Cherries.
37. Assume that chocolate covered cherries come in boxes of 5, 7, and 10. What is the largest
number of chocolate covered cherries that cannot be ordered exactly?
{Hint: If you can get five consecutive amounts of cherries, then you can get all amounts
larger. Here’s why: Suppose you can get the amounts 23,24,25,26,27 , then by the
addition of the box of size 5, you can also get 28,29,30,31,32 , and another addition
of the box of size 5 produces 33,34,35,36,37 and so on. This would also be true of
seven consecutive amounts and ten consecutive amounts, but five consecutive
amounts would occur first. So look for amounts smaller than the first five
consecutive amounts.}
Easy Come, But Not So Easy Go.
38. A man whose end was approaching summoned his sons and said, “Divide my money as I
shall prescribe.” To his eldest son he said, “You are to have $1,000 and a seventh of what
is left.” To his second son he said, “Take $2,000 and a seventh of what remains.” To the
third son he said, “You are to take $3,000 and a seventh of what is left.” Thus he gave each
son $1,000 more than the previous son and a seventh of what remained, and to the last son
all that was left. After following their father’s instructions with care, the sons found that
they had shared their inheritance equally. How many sons were there, and how large was
the estate?
{Hint: Let M be the total value of the estate.
First Son
Second Son
M 1,000
M 1,000
M 3,000
1,000
7
2,000
7
7
Since each son’s share is the same, solve an equation to determine the value of M,
and use it to find the number of sons.}
Round And Round With Sarah And Hillary
39. Sarah and Hillary are racing cars around a track. Sarah can make a complete circuit in 72
seconds, and Hillary completes a circuit in 68 seconds.
a) If they start together at the starting line, how many seconds will it take for Hillary to pass
Sarah at the starting line for the first time.
{Hint: Every time Sarah reaches the starting line must be a multiple of 72 seconds, and
every time Hillary reaches the starting line must be a multiple of 68 seconds. So
they will both be at the starting line at common multiples of 72 and 68.}
b) If they start together, how many laps will Sarah have completed when Hillary has
completed one more lap than Sarah?
{Hint: Let n be the number of laps completed by Sarah, then 72n 68 n 1 .}
Some divisibility rules for positive integers:
1) A positive integer is divisible by 2 if its one’s digit is even.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but 100a 10b c 2 50a 5b c , so if the one’s digit, c is even(divisible
by 2) then the integer abc will also be divisible by 2.
2) A positive integer is divisible by 3 if the sum of its digits is divisible by 3. This process may
be repeated.
Examples: 243 is divisible by 3, but 271 is not.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but 100a 10b c 99a 9b a b c 3 33a 3b a b c , so if the
the sum of the digits
sum of the digits, a b c , is divisible by 3 then the integer abc will also be divisible by 3.
3) A positive integer is divisible by 4 if the ten’s and one’s digits form a two-digit integer
divisible by 4.
Examples: 724 is divisible by 4, but 726 is not.
Here’s why:
Suppose we have the four-digit integer abcd, then its value can be expressed as
1000a 100b 10c d , but
1000a 100b 10c d 1000a 100b 10c d 4 250a 25b 10c d
, so if
the two-digit integer cd
the ten’s and one’s two-digit integer, cd, is divisible by 4 then the integer abcd will also be
divisible by 4.
4) A positive integer is divisible by 5 if its one’s digit is either a 5 or a 0.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but 100a 10b c 5 20a 2b c , so if the one’s digit, c is divisible by 5,
then the integer abc will also be divisible by 5, but the only digits divisible by 5 are 0 and 5.
5) A positive integer is divisible by 6 if it’s both divisible by 2 and divisible by 3.
6) A positive integer is divisible by 7 if when you remove the one’s digit from the integer and
then subtract twice the one’s digit from the new integer, you get an integer divisible by 7.
This process may be repeated.
Examples: 714 is divisible by 7 since 71 – 8 = 63, but 423 is not since 42 – 6 = 36.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but
100a 10b c 90a 9b 3c
10a b 2c
new integer minus twice one's digit
9 10a b 2c 21c 10a b 2c
10
10a b 2c
7 3c
new integer minus twice one's digit
, so if the new integer minus twice the one’s digit, 10a b 2c , is divisible by 7 then so is the
original integer abc.
Or
A positive integer is divisible by 7 if when you remove the one’s digit from the integer and
then subtract nine times the one’s digit from the new integer, you get an integer divisible by
7. This process may be repeated.
Examples: 714 is divisible by 7 since 71 – 36 = 35, but 423 is not since 42 – 27 = 15.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but
100a 10b c 90a 9b 10c
10a b 9c
new integer minus 9 times one's digit
9 10a b 9c 91c 10a b 9c
10
10a b 9c
7 13c
new integer minus 9 times one's digit
, so if the new integer minus nine times the one’s digit, 10a b 9c , is divisible by 7 then so
is the original integer abc.
Or
A positive integer with more than three digits is divisible by 7 if when you split the digits
into groups of three starting from the right and alternately add and subtract these three digit
numbers you get a result which is divisible by 7.
Examples: 1412236 is divisible by 7 since 1 – 412 + 236 = -175, but 130747591 is not since
130 – 747 + 591 = -26.
Here’s why:
Suppose we have the five-digit integer abcde, then its value can be expressed as
10000a 1000b 100c 10d e , but
10000a 1000b 100c 10d e 10010a 1001b 10a b 100c 10d e
integer ab
integer cde
2 100c 10d e 2 10a b
, so if the
7 143 10a b 10a b 100c 10d e
integer cde
integer ab
integer ab minus the integer cde is divisible by 7 then the integer abcde will also be divisible
by 7.
7) A positive integer is divisible by 8 if the hundred’s, ten’s, and one’s digits form a three-digit
integer divisible by 8.
Examples: 1240 is divisible by 8, since 240 is, 3238 is not, since 238 is not even divisible by
4.
Here’s why:
Suppose we have the five-digit integer abcde, then its value can be expressed as
10000a 1000b 100c 10d e , but
10000a 1000b 100c 10d e 10000a 1000b 100c 10d e
8 1250a 125b 100c 10d e , so if the hundred’s,
the three-digit integer cde
ten’s, and one’s three-digit integer, cde, is divisible by 8 then the integer abcde will also be
divisible by 8.
8) A positive integer is divisible by 9 if the sum of its digits is divisible by 9. This process may
be repeated.
Examples: 243 is divisible by 9, but 9996 is not.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but 100a 10b c 99a 9b a b c 9 11a b a b c , so if the
the sum of the digits
sum of the digits, a b c , is divisible by 9 then the integer abc will also be divisible by 9.
9) A positive integer is divisible by 11 if when you remove the one’s digit from the integer and
then subtract the one’s digit from the new integer, you get an integer divisible by 11. This
process may be repeated.
Examples: 1001 is divisible by 11 since 100 – 1 = 99, but 423 is not since 42 – 3 = 38.
Here’s why:
Suppose we have the three-digit integer abc, then its value can be expressed as
100a 10b c , but
100a 10b c 90a 9b 2c 10a b c
new integer minus one's digit
9 10a b c 11c 10a b c
10
10a b c
11c
new integer minus one's digit
, so if the new integer minus the one’s digit, 10a b c , is divisible by 11 then so is the
original integer abc.
Or
A positive integer is divisible by 11 if when you subtract the sum of the ten’s digit and every
other digit to the left from the sum of the one’s digit and every other digit to the left you get a
number divisible by 11. This process may be repeated.
Examples: 9031 is divisible by 11 since 1 0 3 9 11, but 423 is not since
4 3 2 5 .
Here’s why:
Suppose we have the five-digit integer abcde, then its value can be expressed as
10,000a 1,000b 100c 10d e , but
10,000a 1,000b 100c 10d e 9,999 1 a 1,001 1 b 99 1 c 11 1 d e
9,999a 1,001b 99c 11d a c e b d
11 909a 91b 9c d a c e b d
alternating from one's alternating from ten's
, so if the sum of the alternating digits from the one’s digit minus the sum of the alternating
digits from the ten’s digit, a c e b d , is divisible by 11 then so is the original
integer abcde.
Or
A positive integer with more than three digits is divisible by 11 if when you split the digits
into groups of three starting from the right and alternately add and subtract these three digit
numbers you get a result which is divisible by 11.
Examples: 1412290 is divisible by 11 since 1 – 412 + 290 = -121, but 130747591 is not
since 130 – 747 + 591 = -26.
Here’s why:
Suppose we have the five-digit integer abcde, then its value can be expressed as
10000a 1000b 100c 10d e , but
10000a 1000b 100c 10d e 10010a 1001b 10a b 100c 10d e
integer ab
2 100c 10d e 2 10a b
integer cde
, so if the
11 9110a b 10a b 100c 10d e
integer cde
integer ab
integer ab minus the integer cde is divisible by 11 then the integer abcde will also be
divisible by 11.
Divisibility And Conquer
40. Of the following three numbers determine which are divisible by 2, which are divisible by
3, which are divisible by 4, which are divisible by 6, which are divisible by 8, and which are
divisible by 9: 3 10999 736 , 3 10999 534 , 3 10999 952 .
{Hint: Divisibility rules.}
Keep On Dividing And Conquering.
41. Of the following three numbers determine which are divisible by 3, which are divisible by
6, which are divisible by 7, which are divisible by 9, and which are divisible by 11:
1358024680358024679, 864197523864197523, 964197523864197522.
{Hint: Divisibility rules.}
Seven Come Thirteen, Not Eleven.
42. Show that the second divisibility rule for 7 can also be used as a divisibility rule for 13.
Modify the explanation to show why it works for 13.
Cogswell Cogs Or Spacely Sprockets?
43. In a machine, a small gear with 45 teeth is engaged with a large gear with 96 teeth. How
many more revolutions will the smaller gear have made than the larger gear the first time
the two gears are in their starting position?
{Hint: A revolution of the smaller gear is a multiple of 45 teeth, and a revolution of the
larger gear is a multiple of 96 teeth. So the gears are again in the starting positions
at common multiples of 45 and 96.}
My Tile Cutter Is Broken, So What Now?
44. The figure shows that twenty-four 8"12" rectangular tiles can be used to tile a 48" 48"
square without cutting tiles.
a) Is there a smaller sized square that can be tiled without cutting using 8"12" tiles? If so,
find it.
{Hint: The dimension of the square tile must be a multiple of both dimensions of the
rectangular tiles.}
b) What is the smallest square that can be tiled without cutting using 9"12" tiles?
{Hint: See the previous hint.}
Solving Without Completely Solving
45. If a b 2 and a b 3 , then find a 4 b 4 .
2
2
{Hint: Square the two equations.}
Odds, Evens, What’s The Difference?
46. What do you get if the sum of the first 8,000,000,000 positive odd integers is subtracted
from the sum of the first 8,000,000,000 positive even integers?
{Hint: 2 4 6 8
16,000,000,000 1 3 5 7
15,999,999,999 }
The Great Luggage Caper
47. Great Aunt Christine is going for her annual holiday to Barbados. She sends her butler John
down to the airport with her collection of suitcases, each of which weighs either 18 or 84
pounds, and is informed that the total weight checked-in is 652 pounds. Show that this is
impossible without listing and checking all the possible combinations of 18 and 84 pound
suitcases.
{Hint: The total weight of the suitcases is of the form 18x 84 y , where x and y are
nonnegative integers. So the expression must be divisible by the greatest common
factor of 18 and 84.}
Gerry Benzel’s Favorite Problem
48. A bottle and a cork together cost $1.10. If the bottle costs $1.00 more than the cork, what
does the cork cost?
{Hint: Let x be the cost of the cork and y the cost of the bottle.}
Getting Solutions Without Actually Solving
49. Notice that
x a x b x 2 a b x ab
x a x b x c x3 a b c x 2 ab ac bc x abc
x a x b x c x d x 4 a b c d x3 ab ac ad bc bd cd x 2
abc abd acd bcd x abcd .
a) Use inductive reasoning to determine the value of the coefficient of x n1 and the constant
term in the expansion of the following product: x a1 x a2 x an .
b) Use the previous result to determine the sum of the seventeenth powers of the 17
solutions of the equation x17 3x 1 0 .
{Hint: The Fundamental Theorem of Algebra guarantees that the equation
x17 3x 1 0 has seventeen solutions(counting duplicates). The seventeen
solutions of x a1 x a2 x a17 x17 3x 1 0 are a1 , a2 , , a17 . So
adding the seventeen equations together yields:
a117 3a1 1 0
a17
2 3a2 1 0
. Now use the previous result.}
17
a17
3a17 1 0
a
17
1
a17
2
17
a17
3 a1 a2
a17 17 0
Dots And Dashes, But It’s Not Morse Code.
50. The diagram shows a sequence of shapes T1 , T2 , T3 , T4 , . Each shape consists of a number
of squares. A dot is placed at each point where there is a corner of one or more squares.
T1
Shape
T1
T2
T3
T4
T2
number of rows, n
1
2
3
4
T3
number of squares, S
1
4
9
16
T4
number of dots, D
4
10
18
28
a) Use inductive reasoning to find a formula for S in terms of n.
b) How many squares are in shape T25 ?
c) Use inductive reasoning to find a formula for D in terms of n.
{Hint: Notice that D n2 in the last column is always a multiple of 3.}
d) How many dots are in shape T25 ?
It Squares; It cubes; It does it all!
51. If a b 10 and a b 5 , then what is the value of a 2 ab b2 ?
3
3
{Hint: a b a 2 ab b 2 a 3 b3 .}
D n2
3
6
9
12
52.
Two Squares Don’t Get Along – A Difference Of Squares!
If x
y 20,000 5 , x10,000 y10,000 4 , x5,000 y5,000 3 , x 2,500 y 2,500 2 ,
x 2,500 y 2,500 1 , then find the value of x 40,000 y 40,000 .
20,000
and
Hint: a b a b a 2 b 2 .
How Touching!
53. Four circles, each of which has a diameter of 2 feet, touch as shown. Find the area of the
shaded portion.
{Hint: The area of the shaded portion would be the area of the square minus the area of the
four circular sectors.
}
The method of finite differences can be used to produce formulas for lists of numbers. For
example, if the list of numbers is 5,7,9,11,13,…, then the list of first finite differences is the list
of differences of consecutive numbers: 2 , 2 , 2 , 2 ,
or more simply, 2,2,2,2,….
75 97 119 1311
Whenever the list of first finite differences is a repetition of the same number, it means that the
original list of numbers can be produced by a formula of the form an b , where n represents
the position in the list. Here’s why:
original
first finite differences
ab
a
2a b
a
3a b
a
4a b
In fact, the repeated number in the first finite differences will always be the coefficient of n.
In our example, it must be that a 2 and a b 5 . So we get that a 2 and b 3 , and a
formula for the list of numbers 5,7,9,11,13,… is 2n 3. Sometimes the list of first differences
is not a repetition of the same number. An example of this is the list 3,9,19,33,51,73,…. The
list of first finite differences is 6,10,14,18,22,…. As you can see it’s not a repetition of the
same number. Now let’s calculate the list of second finite differences: 4,4,4,4,…. Whenever
the list of second finite differences is a repetition of the same number it means that the original
list of numbers can be produced by a formula of the form an2 bn c , where again n
represents the position in the list. Here’s why:
original
first finite differences
second finite differences
a b c 4a 2b c
3a b
5a b
2a
2a
9a 3b c 16a 4b c
7a b
9a b
2a
25a 5b c
In fact, the repeated number in the second finite differences will always be twice the coefficient
of n 2 . In our example, it must be that a 2 , a b c 3 , and 4a 2b c 9 . Substituting the
b c 1
value of a into the next two equations leads to the system
. Subtracting the first
2b c 1
equation from the second equation leads to b 0 , so the values are a 2 , b 0 , and c 1 . So
a formula for the list of numbers 3,9,19,33,51,73,… is 2n 2 1. Similar results hold for higher
order differences which are repetitions of the same number. You may use the method of finite
differences to solve the following two problems.
What’s the Diff?
54. Find a formula for each of the following lists of numbers:
a) 2,5,8,11,…
b) 32 ,2, 52 ,3, 72 ,
c) 4,7,12,19,28,
d) 3,7,13,21,31,43
e) 1, 1 2 , 1 2 3 , 1 2 3 4 , 1 2 3 4 5 ,
f) 1, 1 3 , 1 3 5 , 1 3 5 7 , 1 3 5 7 9 ,
g) 1,15,53,127,249,…
h) 1, 1 4 , 1 4 9 , 1 4 9 16 , 1 4 9 16 25 , 1 4 9 16 25 36 ,
You Want Me To Cut The Pizza Into How Many Slices?
55. A chord is a line segment joining two points on a circle. Here, n is the number of chords.
n 1
n2
n3
n4
n5
Associated with each number of chords, n, is the maximum number of regions that the circle is
divided into by the n chords. For the first five numbers of chords, the list of the maximum
number of regions is 2,4,7,11,16 . Use the method of finite differences to find a formula for the
maximum number of regions that a circle can be divided into using n chords, and use the
formula to predict the maximum number of regions a circle can be divided into using 60 chords.
Now What Was That Last Digit?
56. Peggy is writing the numbers 1 to 9,999. She stops to rest after she has written a total of
1,000 digits. What is the last digit that she wrote?
{Hint:
1 digit numbers(1-9)
2 digit numbers(10-99)
3 digit numbers(100-999)
4 digit numbers(1,000-9,999)
Quantity
9
90
900
9000
Total number of digits
9
180
2,700
36,000
}
On The Mark, Off The Mark, Or Bull’s Eye?
57. In the following figure, the curves are concentric circles with the indicated radii. Which
shaded region has the larger area, the inner circle or the outer ring?
4
3
5
Calculate the area of each region and check your visual estimation ability.
I Hate This Problem To The Nth Degree.
58. Use the following properties of exponents to find the exact value of the given expressions.
xn xm x nm , xy x n y n , x n x nm
n
390,000 690,000
a) 89,999 90,000
2
9
m
2100,000 3110,000
b) 100,000 5000
6
9
We use a base 10 number system. A number written with the digits abcd actually represents the
number a 103 b 102 c 10 d , where the digits a, b, c, and d can be any of the numbers 0, 1,
2, 3, 4, 5, 6, 7, 8, or 9. If we want to use a base of 2, then abcd would represent the number
a 23 b 22 c 2 d , where the digits a, b, c, and d can be the numbers 0 or 1.
I’m Thinking Of A Number From 1 To 31.
59. A person is asked to think of a whole number from 1 to 31. The person is shown the
following 5 cards and asked to identify the cards that contain the number he is thinking of.
1 3 5 7
2
3 6 7
9 11 13 15
10 11 14 15
12 13 14 15
17 19 21 23
18 19 22 23
20 21 22 23
25 27 29 31 26 27 30 31
8 9 10 11 16 17 18 19
28 29 30 31
12 13 14 15
20 21 22 23
24 25 26 27
24 25 26 27
28 29 30 31
28 29 30 31
4
5 6 7
By adding the numbers in the upper left corners of the identified cards, you can always
determine the person’s number. Use base 2 numbers to explain how the trick works.
Oh Yeah, Well I’m thinking Of A Number From 1 to 63.
60. See if you can extend the previous trick to work for whole numbers from 1 to 63 using 6
cards instead of 5. Complete the numbers on the 6 cards in order for the trick to work.
1 3 5 7 9 11 13 15
2
17 19 21 23 25 27 29 31
33 35 37 39 41 43 45 47
49 51 53 55 57 59 61 63
4
16
8
32
Call it like you see it.
61. Consider the following figure:
a) What fraction of the large square is shaded
?
b) What fraction of the large square is shaded
?
c) What fraction of the large square is shaded
?
Let’s Not Go Overboard.
62. This problem in one form or another has been around for nearly 2,000 years – if not longer.
Josephus, a Jewish historian of the first century A. D., mentions it. In all its forms the
problem seems to have a violent streak. Here’s our version of it: One-hundred sailors are
arranged around the edge of their ship. They hold in order the numbers from 1 to 100.
Starting the count with number 1, every other sailor is pushed overboard into the cold North
Atlantic waters until there is only one left – the survivor.
a) What number do you want to be holding in order to be the survivor?
b) How many times will the survivor be skipped during this process?
c) Find the number of the last sailor to be pushed overboard.
d) Find the number of the second to last sailor to be pushed overboard.
{Hint: Consider the problem for smaller numbers of sailors and work up: For example
with 10 sailors you have
10
1
1
2
3
9
first
elimination
8
7
3
9
second
elimination
4
5
5
7
6
1
9
third
elimination
the
survivor
5
5
The survivor, 5, is skipped three times. The last sailor to be pushed overboard is
9, and the next to last sailor pushed overboard is 1.}
A Springtime Path
63. Determine the number of different paths for spelling the word APRIL:
A
P
R
I
L
P
R
I
L
R
I
L
{Hint: The letters essentially form a tree diagram.}
I
L
L
A Checkered Present
64. a)How many squares can you find on a 8 8 checkerboard?
b) an n n checkerboard?
Hint: Start with smaller boards and look for a pattern.
1 large and 4 small
1+4=5
1 large, 4 medium, and 9 small
1 + 4 + 9 = 14
1 extra large, 4 large, 9 medium, and 16 small
1 + 4 + 9 + 16 = 30
Put A Sock In It Brad And Angelina.
65. Mr. Smith left on a trip very early one morning. Not wishing to wake Mrs. Smith, Mr.
Smith packed in the dark. He had socks that were alike except for color, and his socks
came in six different colors. Find the least number of socks he would have had to pack to
be guaranteed of getting
a) at least one matching pair of socks.
b) at least two matching pairs of socks.
c) at least three matching pairs of socks.
d) at least four matching pairs of socks.
{Hint: He could actually pack as many as 6 socks and still not have a matching pair.
1
Color 1
1
Color 2
1
Color 3
1
Color 4
1
Color 5
1
Color 6
}
Facebook Shmacebook.
66. After graduation exercises, each senior gave a snapshot of himself or herself to every other
senior and received a snapshot in return. If 2,000,810 snapshots were exchanged, how
many seniors were in the graduation class?
Hint: Start with small classes and look for a pattern.
It All Adds Up To Something.
67. a) There are 120 five-digit numbers that use all the digits 1 through 5 exactly once. What is
the sum of the 120 numbers?
12345
12354
Hint:
120 numbers How many of each digit occur in each column?
54321
b) If the digits can be repeated, then there are 3,125 five-digit numbers that can be formed.
What is the sum of the 3,125 numbers?
Consider the sets A and B inside a universal set U. n U n A
B n A
B , so we get that
n U n A n B n A B n A B . This rearranges into
n A
B n A n B n U n A
B , and since n A B 0 , it must be that
n A B n A n B n U . This means that the number of elements in the intersection of
A and B is at least n A n B n U , and it also means that if n A n B n U 0 , then
it’s possible that n A B 0 . This result can be extended to the case of three sets as follows:
n A B C n A B C n A n B C n U n A n B n C n U n U
so n A B C n A n B n C 2 n U . It can further be extended to the case of four
sets as follows:
n A B C D n A B C D n A n B C D n U
,
so
n A n B n C n D 2 n U n U
n A B C D n A n B n C n D 3 n U .
In general, you can show that
n A1 A2
Ak n A1 n A2 n Ak k 1 n U .
Also, n A
B n A and n A B n B , so n A B min n A , n B . In general,
you can show that n A1
A2
Ak min n A1 , n A2 ,
, n Ak
War Is Hell!
68. In a group of 100 war veterans, if 70 have lost an eye, 75 an ear, 80 an arm, and 85 a leg:
a) at least how many have lost all four?
{Hint: See the previous discussion.}
b) at most how many have lost all four?
Not All Things Must Pass.
69. Forty-one students each took three exams: one in Algebra, one in Biology, and one in
Chemistry. Here are the results
12 failed the Algebra exam
5 failed the Biology exam
8 failed the Chemistry exam
2 failed Algebra and Biology
6 failed Algebra and Chemistry
3 failed Biology and Chemistry
1 failed all three
How many students passed exams in all three subjects?
{Hint: Make a Venn diagram.}
Man Or Woman, We Mean Business.
70. Out of 35 students in a math class, 22 are male, 19 are business majors, 27 are first-year
students, 14 are male business students, 17 are male first-year students, 15 are first-year
students who are business majors, and 11 are male first-year business majors.
a) How many upper class female non-business majors are in the class?
b) How many female business majors are in the class?
Female
Male
Business
First-year student
Non-business
If You Can’t Work On Transmissions, That’s The Brakes.
71. A car shop has 12 mechanics, of whom 8 can work on transmissions and 7 can work on
brakes.
a) What is the minimum number who can do both?
b) What is the maximum number who can do both?
c) What is the minimum number who can do neither?
d) What is the maximum number who can do neither?
{Hint: See the hint for problem #68.}
Trains, Planes, And Automobiles
72. 85 travelers were questioned about the method of transport they used on a particular day.
Each of them used one or more of the methods shown in the Venn diagram. Of those
questioned, 6 traveled by bus and train only, 2 by train and car only, and 7 by bus, train, and
car. The number x who traveled by bus only was equal to the number who traveled by bus
and car only. 35 people used buses, and 25 people used trains. Find:
Train
Bus
6
x
7
x
Car
2
U
a) the value of x.
b) the number who traveled by train only.
c) the number who traveled by at least two methods of transport.
d) the number who traveled by car only.
Hardback Or Paperback Writer?
73. Books were sold at a school book fair. Each book sold was either fiction or nonfiction and
was either hardback or paperback. The chair-person of the book-selling committee can’t
remember exactly how many hardback books of fiction were sold, but he does remember
that
30 books were sold in all
20 hardcover books were sold
15 books of fiction were sold
a) What is the smallest possible number of hardback books of fiction sold?
b) What is the largest possible number of hardback books of fiction sold?
{Hint: See the hint for problem #68.}
It’s Just Gotta Be 3.
74. a) Show that for any integer n, 3 divides n3 n .
{Hint: n3 n n n 2 1 n 1 n n 1 }
b) Show that for any integer n, 3 divides n5 n .
Even So, It’s Odd.
75. Show that for every positive integer n, n2 3n 8 is even.
{Hint: n is either even or odd, so take it from here.}
Logarithms Are Very Overrated.
76. If 3 4 , 4 5 , 5 6 , 6d 7 , 7e 8 , and 8 f 9 , then what is the value of the product
abcdef ?
{Hint: From 3a 4 and 4b 5 , you can conclude that 3ab 5 . So just keep going.}
a
b
c
Given The Least, What’s The Greatest?
77. Given that 12 is the least common multiple of the positive integers 66 , 610 , and k . How
many different values of k are possible?
10
Real-valued Function, What’s Your Function?
78. Suppose that f is a real-valued function with f x 2 f 2002
Find
x 3x for all x 0 .
f 2 .
{Hint: Plug in 2 and 1001, and solve for f 2 .}
An Odd Product.
3 5 7 9
4021
79. Find the value of the following product:
.
1 1 3 5
4017
{Hint: What factors can you cancel out?}
An Even Sum.
80. Find the one’s digit of the following sum: 9 81 729 92010 .
{Hint: Start small, and look for a pattern.}
Don’t Put All Of Your Eggs In One Basket!
81. If eggs are taken from a basket two at a time, then one egg remains in the basket. If eggs are
taken three at a time from the same basket, then two eggs remain in the basket. If eggs are
taken four at a time from the same basket, then three eggs remain in the basket. If eggs are
taken five at a time from the same basket, then four eggs remain in the basket. If eggs are
taken six at a time from the same basket, then five eggs remain in the basket. If eggs are
taken seven at a time from the same basket, then no eggs remain in the basket. What is the
fewest possible number of eggs in the basket?
{Hint: If N is the number of eggs in the basket, then N 1 must be a multiple of 2, N 1
must be a multiple of 3, N 1 must be a multiple of 4, N 1 must be a multiple of 5, and
N 1 must be a multiple of 6. So N 1 must be a multiple of the LCM of 2, 3, 4, 5, and 6.
Also, N must be a multiple of 7.}
Don’t Be So Irrational.
82. Show that log 2 3 is an irrational number.
a
a
. Then 3 2 b . Raise both
b
sides to the b power, and use the Fundamental Theorem of Arithmetic.}
{Hint: Suppose that log 2 3 is a rational number, i. e. log 2 3
Given The Greatest, What’s The Least?
83. If the product of two natural numbers is 273852711 , and their greatest common factor is
23345 , then what is their least common multiple?
The A, B, C, And D Of Education
84. The results of a survey were the following:
12 students take art, 20 take biology, 20 take chemistry, 8 take drama, 5 take art and
biology, 7 take art and chemistry, 4 take art and drama, 16 take biology and chemistry, 4
take biology and drama, 3 take chemistry and drama, 3 take art, biology, and chemistry, 2
take art, biology, and drama, 2 take biology, chemistry, and drama, 3 take art, chemistry,
and drama, 2 take all four, 71 take none of the four
Chemistry
Biology
Drama
Art
U
a) How many students participated in the survey?
b) How many take exactly one class?
c) How many take exactly two classes?
d) How many take exactly three classes?
e) How many take two or more classes?
The Truth About Cats And Dogs And Birds And Fish.
85. A survey of 136 pet owners resulted in the following information: 49 own fish; 55 own a
bird; 50 own a cat; 68 own a dog; 2 own all four; 11 own only fish; 14 own only a bird; 10
own fish and a bird; 21 own fish and a cat; 26 own a bird and a dog; 27 own a cat and a
dog; 3 own fish, a bird, a cat, but no dog; 1 owns fish, a bird, a dog, but no cat; 9 own fish,
a cat, a dog, but no bird; and 10 own a bird, a cat, a dog, but no fish.
Bird
Cat
Dog
Fish
U
How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs?
Lucky 13, I Repeat, Lucky 13.
86. Show that every six-digit number of the form abc,abc (for example 281,281 or 435,435) is
divisible by 13.
The Euclidean algorithm can be used to find the greatest common factor of two whole numbers.
It can also be used to express the greatest common factor as a combination of the two whole
numbers. Here’s an example:
Let’s find the gcf of 738 and 621 using the Euclidean algorithm, and express it as a combination
of 738 and 621.
738 1 621 117
621 5 117 36
117 3 36 9
36 4 9
So the gcf of 738 and 621 is 9. Now work backwards through the equations.
9 117 3 36 117 3 621 5 117 16 117 3 621 16 738 621 3 621
16 738 19 621.
So 16 738 19 621 9 .
The Greatest Common Combination.
87. Find the greatest common factor of the following pairs of numbers, and express it as a
combination of the two numbers.
a) 24 and 54
b) 72 and 160
c) 5291 and 11951
The greatest common factor of 7 and 9 is 1. The equation 7 5 9 4 1, expresses 1 as a
combination of the numbers 7 and 9. The equation also gives a way of solving the following
problem:
You have an unlimited supply of water, a drain, a large container, and two jugs which can hold
7 gallons and 9 gallons, respectively. How can you manage to end up with exactly 1 gallon of
water in the large container?
The equation tells you how to do it. Add four of the 9 gallon containers of water into the large
container, and then you remove 5 of the 7 gallon containers of water from the large container.
You’ll be left with 1 gallon of water in the large container.
Forget About Your Whiskey, Can You Hold Your Water?
88. Solve the previous water problem, except this time you have:
a) 36 gallon and 60 gallon jugs, and you want 12 gallons in the large container.
b) 40 gallon and 70 gallon jugs, and you want 10 gallons in the large container.
c) 450 gallon and 750 gallon jugs, and you want 150 gallons in the large container.
I’m Just Trying To Convert You To The Same Religion.
89. a) Find a linear function y mx b that converts the arithmetic sequence 3, 1,1,3,5,
into the arithmetic sequence 2, 12, 22, 32, 42, .
{In other words, you want to find values of m and
2 3m b, 12 m b, 22 m b, 32 3m b, 42 5m b, .}
b
so
that
b) Suppose you are given two arithmetic sequences a, a h, a 2h, a 3h, a 4h,
and
c, c k , c 2k , c 3k , c 4k , . Is it always possible to find a linear function
y mx b that converts the first sequence into the second sequence?
{In other words, is it always possible to find values of m and b so that
c am b, c k a h m b, c 2k a 2h m b , c 3k a 3h m b , ?}
c) The graph of the linear function y 157 x 13 passes through two points with integer
coordinates: 20, 9 and 10, 5 . Are there other points on the graph with integer
coordinates? If so, how many?
{Think about the two arithmetic sequences 20,10, 40,70,100,
and 9,5,19,33, 47,
}
d) If it is known that the graph of y mx b passes through two points with integer
coordinates, must there be other points on the graph with integer coordinates? If so, how
many?
e) The graph of the linear function y x 12 does not pass through any point with integer
coordinates. Here’s why: If y x 12 , and both x and y are integers, then y x 12 .
But this would mean that 12 is an integer. So it’s possible for the graph of a linear
function to not contain any points with integer coordinates. Is it possible for the graph
of a linear function to contain just one point with integer coordinates?
{Consider the linear function y 2 x .}
Just your average airplane flight.
90. An airplane travels between two cities. The first half of the distance between the two cities
is travelled at an average speed of 600 mph. The second half of the distance is travelled at
an average speed of 900 mph. Find the average speed of the airplane for the entire trip.
distance travelled
{Hint: average speed
.}
time of travel
Don’t Go Rushin’ Through Your Division Problems.
91. There is a method of division that is similar to Russian Peasant Multiplication that uses
doubling and subtraction. Here’s how it works:
For 139 12
Keep doubling 12 until the next doubling would exceed 139.
12 112
24 2 12
48 4 12
96 8 12
Now subtract these values from largest to smallest if possible from 139 until you are left
with a non-negative value less than 12.
139 96 43 , 43 24 19 , 19 12 7 . So
139 12 8 2 1 r 7
11r 7
. Apply this method to
evaluate 185 8 and 656 58 .
Sometimes Reduction Can Get In The Way Of Induction.
92. Observe that
1 3
1 2
2 4
1
1 4
1
1
2
2
2 3 6
1
1
1 5
1 2 1 2 1 2
2 3 4 8
1
1
1
1 6
1
1
1
1
2
2
2
2
2 3 4 5 10
.
Use inductive reasoning to make a conjecture about the value of
1
1
1
1
1 2 1 2 1 2 1 2 .
2 3 4 n
1
1
1
1
Use your conjecture to find the value of 1 2 1 2 1 2 1
.
2
2
3
4
1,
000,
000
1
1
1
If this process were to continue indefinitely, i.e. 1 2 1 2 1 2 , what would be the
2 3 4
resulting value?
In College Algebra, you learned something called the Remainder Theorem for evaluating
polynomials. It says that if you want to find p c where p x is a polynomial, then you can
divide p x by x c , and the remainder will be p c . You also learned synthetic division,
which gave you an efficient way to divide p x by x c , and therefore an efficient way to
evaluate p c .
As an example, suppose you want to find p 3 , where
p x 2 x 3 x 2 4 x 5 . To find the remainder when p x is divided by x 3 , we’ll use
synthetic division:
3 2 1 4 5
6
21 51
2 7
17 56
b
numeral
So p 3 56 .
When
you
dnb dn1b
n
n1
are
given
the
base
d n d n1d n2
d1d0 ,
its
value
is
d1b d0 which is what you get when you find p b for the polynomial
p x d n x n d n1 x n1 d1 x d 0 . So an efficient method of converting the base b numeral
d n d n1d n2 d1d0 into decimal is to use synthetic division.
b dn
d n1
d0
bdn
dn
This approach is called Horner’s Method.
?
Little Jack Horner Sat In A Corner Converting His Base B Numerals.
93. Convert the following numerals into equivalent decimals using Horner’s Method:
a) 11010112
b) 120221103
c) 621347
There is an efficient method for converting base ten(decimal) fractions into base 2. As an
example, let’s convert the decimal fraction .8215 into its base 2 equivalent.
.8125 2 1.625
.625 2 1.25
decimal part
.25 2 0.5
decimal part
.5
decimal part
2 1.0
Whole number part
1
1
0
1
The process stops when the decimal part is zero. So .8125 .11012 .
Double, Double, Less Toil, And No Trouble.
94. Use the previous algorithm to convert the following base 10 numerals into base 2.
a) 1.6875
b) 2.46875
c) .4
Are You Ready For Prime Time?
95. If you add up all the prime numbers less than one-million, will it be an even number or an
odd number? Why?
Cheaper By The Dozen?
96. Find the sum of the divisors of 6480 that are multiples of 12.
What Four?
97. How many four-digit numbers are multiples of 15, 20, and 25?
