Download Unit 1 - nsmithcac

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Blue Problems Prime Factorization, Greatest Common Factor and Simplifying Fractions
1. The number n is a prime number between 20 and 30. If you divide n by 8, the
remainder is 5. What is the value of n?
2. Five consecutive two-digit positive integers, each less than 30, are not prime. What
is the largest of these five integers?
3. What is the sum of the three distinct prime factors of 47,432?
4. What is the tens digit of the product of the first six prime numbers?
5. What is the greatest prime factor of 3105?
6. In a certain code, each of the 26 letters of the English alphabet is represented by
a number (A=1, B=2, C=3,... Z=26). A word is then encoded by multiplying the
numbers that represent its letters. For example, CAT is encoded by 3* 1* 20 = 60,
MATH is encoded by 13*1*20*8 = 2080.
Find a word that would be encoded as 7560 and explain how you found it. Could
there be other words? Explain why or why not.
7. Apples. At harvest time, the orchards of Mr. MacIntosh, Mr. Jonathan, and Mr. Delicious
had yielded 314,827 apples, 1,199,533 apples, and 683,786 apples, respectively. While having
lunch with Jonathan the following Sunday, MacIntosh mentioned the number of apples he
would have left over if he divided his harvest equally among all the apple dealers.
“Why don’t you sell those extra apples to me,” suggested Jonathan, “and then I’ll be able to
divide my apples equally among all the dealers.”
“Sorry, said MacIntosh, “but Mr. Delicious made the same suggestion for the same reason,
and I’ve already accepted his offer.” How many apple dealers are there?
8. What is the positive difference between the two largest prime factors of 159,137?
9. Last Prime Date. If both the month and the day are prime numbers, then consider
this date to be a prime date. For example, July 19 (7/19) is a prime date. What are
the first and last prime dates of the year?
10. What number can be subtracted from both the numerator and denominator of
19/24 so that the resulting fraction will be equivalent to 3/4?
11. What fraction of the eleven letters in the word “MISSISSIPPI” are I’s? Express
your answer as a common fraction.
12. The fraction x does not change its value when 3 is added to both its numerator
5
and its denominator. What is the value of x?
13.
A fraction is equivalent to 3/5. Its denominator is 60 more than its numerator.
What is the numerator of this fraction?
14. Find two pairs of variable expressions that have 24xy2 as their greatest common
factor.
15. The sum of the numerator and denominator of fraction is 160. The fraction is
equivalent to 3 . What is the original fraction?
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Blue Solutions
1. The primes between 20 and 30 are 23 and 29. If we divide 23 by 8, we get a remainder of 7. If we
divided 29 by 8, we get a remainder of 5. The value of n must be 29.
2. This problem is asking about two-digit numbers that are not prime, in other words, they will each be
composite numbers. We are given an upper limit of 30, so all of these numbers will be in the teens or
twenties. Let’s go ahead and list them out, circling the numbers that are not prime:
10 11 12 13 14
15
16 17 18
19 20 21
22 23 24 25 26 27 28 29
5 non-primes
The largest of the five consecutive non-prime integers is 28.
3. The prime factorization of 47,432 is 23x72x112. The sum of the three prime factors is 2 + 7 + 11 = 20.
4. The first six prime numbers are 2, 3, 5, 7, 11 and 13. The product of the three primes 7, 11 and 13 is
the very special number 1001. This is helpful in determining the product of all the numbers. Since 2  3
 5 = 30 and 7  11  13 = 1001, then the product of all six primes is 30  1001 = 30,030. The tens digit
is 3.
5. Since the digits of 3105 have a sum of 3 + 1 + 0 + 5 = 9, 3105 is divisible by 9. In fact, 3105 = 3 2 x 345.
However, the digits of 345 have a sum of 3 + 4 + 5 = 12, so 345 must be divisible by 3. This implies
that 3105 = 33 x 115. Finally, since 115 ends in 5, it is divisible by 5, giving 3105 = 3 3 x 5 x 23. The
greatest prime factor of 3105 is 23.
6. LINE, ALIEN, REGAL, BORN, LARGE, BARON are only some of the many.
We began to solve the equation by making a factor tree. We factored the number 7560 until we
received four numbers that translated into letters, which we formed into a real word. This is an
example of our factor tree:
Numbers used:
Translation of numbers:
Letters unscrambled:
7560
/\
20 378
/\ / \
4 5 18 21
DE RU
RU DE
7. Apples. The number of dealers must be a divisor of 314,827 + 1,199,533 = 1.514.360, and of 314,827 +
683,786 = 998,613. The greatest common divisor of these two numbers is 131. Since 131 is a prime,
there are 131 dealers.
8. Prime factoring a number is arguably the most difficult thing to do in mathematics. There’s no quick
way to do it. But, the divisibility rules will show that no prime less than 11 will divide the number in the
problem. Using the divisibility rules is a good way to get started.
The divisibility rule for 2 says that a number is divisible by 2 if the last digit is even. The last digit is
7, so the number is not divisible by 2.
The divisibility rule for 3 says that a number is divisible by 3 if the sum of the digits is divisible by 3.
Since 1 + 5 + 9 + 1 + 3 + 7 = 26 is not a multiple of 3, then 159,137 is not divisible by 3, either.
A number divisible by 5 has units digit 0 or 5, so 159,137 is not divisible by 5.
The rule for divisibility by 7 is tricky. In short, remove the last digit double it and subtract that from
the number remaining; repeat until a number is reached that can identified as divisible or not divisible
by 7. For 159,137; 15,913 – 2(7) = 15,899; then 1589 – 2(9) = 1571; then 157 – 2(1) = 155; and, finally,
15 – 2(5) = 5, which is not divisible by 7, so 159,137 is not divisible by either.
I determine of a number is divisible by 11, alternately add and subtract the digits. For this number, 1 –
5 + 9 – 1 + 3 – 7 = 0, which is a multiple of 11, so the number is divisible by 11.
In fact, 159,137 = 11  14,467. Then, a calculator can be used to show that 14,467 = 17  23  37. The
difference between the two greatest factors is 37 – 23 = 14.
9. Last Prime Date. 2/2 and 11/29/ The smallest prime is 2, so 2/2 would be the first prime date. The
largest prime less than or equal to 12 is 11. November, the eleventh month, has 30 days. The largest
prime less than or equal to 30 us 29.
10. Generate fractions by subtracting 1, 2, 3, …, from the numerator and denominator of given fraction
19/24. Stop when the result is a fraction equivalent to ¾. Subtracting 1 gives 18/23. Subtracting 2
gives 17/22, subtracting 3 gives 16/21, subtracting 4 gives 15/20 = 3/4. The number 4 was subtracted
from the numerator and denominator.
11. There are 4 I’s in the eleven-letter word MISSISSIPPI, so 4/11 of the word’s letters are I’s.
12. If x is 5, we have 5/5 = 1 before adding 3’s to numerator and denominator and 8/8 = 1 afterward. To
find the algebraically, we could set up the equation x = x + 3 and solve for x.
5 5+ 3
3
13. In the simplified fraction , the numerator is 2 less than the denominator. In the equivalent fraction,
5
the numerator 60 less than the denominator. This means that 2 units in the simplified fraction equal
60 units in the equivalent fraction, so 3 units would equal 90. The equivalent fraction must be 90 ,
150
and the numerator is 90.
14. Sample answers: 48xy2z and 72x2y3, 24x5y3z and 96xy2.
15. Let’s assume we do not have any idea where to start. We know we need a fraction equivalent to 3 .
7
6
Multiplying the numerator and denominator by 2 gives us the equivalent fraction
, but the sum of
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the numerator and denominator is only 20. The next fraction is 9 , but again its sum of 30 falls way
21
12
short of 160. The next fraction is
, and gives a sum of 40. We are still far from the sum of 160 we
28
need, but notice that our sum is increasing by 10 each time we increase the factor by which we are
multiplying the numerator and denominator. We just multiplied both by 4. We need the sum to go up
another 160 – 40 = 120, which is 12 more 10s. Raising our factor of 4 by 12, we get 16. Let’s try
multiplying the numerator and denominator by 16. We get 48 , which has the correct sum of 160.
112
Rather than using the above Guess, Check & Revise method, we could set up the situation
algebraically. We know we have to multiply the numerator and denominator each by the same number,
x, in order to get an equivalent fraction. We also know the hew numerator and denominator must add
to 160, so 3x + 7x = 160 or 10x = 160 or x = 16. We have determined we must multiply the numerator
and denominator by 16, and this yields 48 . Can we see now why the sums in our first solution kept
112
increasing by 10?
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources were not known.
Problems
Bibliography Information
6
The Math Forum @ Drexel
(http://mathforum.org/)
1 – 5, 8 , 10 – 13 ,
15
Math Counts
(http://mathcounts.org)
7
Friedland, Aaron J. Puzzles In
Logic & Math. New York City:
Dover Publication, 1970.
9
Collier, C. Patrick. Menu
Collection Problems Adapted
from Mathematics Teaching in
the Middle School. New York:
National Council of Teachers of
Mathematics, 2000. Print.
14
Larson, Boswell, Kanold, and
Stiff. Mathematics Concepts
and Skills Course 2 Math Log.
McDougal Littell, 2001.