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Math 41 – Practice for Test 4
1.
In a survey, 25 out of 30 fourth graders said they liked their music class. The rest said they did not.
a) Among those surveyed, what is the ratio of students who like their music class to those who do
not?
b) Among those surveyed there are _____ times as many who like their music class as those who
do not.
c) Among those surveyed there are _____ times as many who do not like their music class as
those who do.
d) What fraction of the students interviewed like their music class?
e) What percent of the students interviewed don’t like their music class?
2.
The ages of a mother and a daughter are 50 years and 20 years.
a) Make an additive comparison of their ages
b) Make a multiplicative comparison of their ages.
3.
A mathematician tells you that the ratio of museums to parks in a town is 0. What would you
conclude?
4.
Perry is offered two jobs. Job #1 pays $500 for a 40-hour week while Job #2 pays $450 for a 36hour week. Which job has a better pay rate? Explain how you decide.
5.
Meg paid $12 for ¾ pound of crab. What is the price per pound?
a) Show how to answer this question using a part-whole diagram.
b) Show how to answer this question using a proportion.
6.
Two college students, Abe and George, were hired to paint the floor in the gym. Abe, who worked
more slowly, painted only 3/5 as much of the floor as George did.
a) Mark the drawing of the floor to show the amounts of floor painted by each of the students.
b)
c)
d)
e)
Abe’s part is how many times as much as George’s part?
What part of the floor did Abe paint?
What is the ratio of the part painted by George to the part painted by Abe?
If they are paid $500 to paint the floor, what would be a fair split of the $500?
7.
At a picnic, one table seating 10 people has 3 platters of food. A second table seating 14 people
has 4 platters of food. Which table is allotting more food per person? Explain your answer using
fractions rather than decimals.
8.
Use a rectangle to represent a submarine sandwich. Divide the sandwich into two sections A and B
so that the ratio of A to B is 3:4.
a) What fraction of the whole sandwich is section A?
b) Section B is how many times as long as section A?
c) Section A is how many times as long as section B?
1
9.
One day, Kai and Mai are discussing their ages. They were born on the same day but not the same
year. Kai says, “Today I’m 5 years older than you are.” Mai says, “Yes, and 4 years ago you were
50% older than I was then.” What will be the ratio of Mai’s age to Kai’s age, 6 years from today?
Show your work.
10. Three cities, Avalon, Bayside and Clarkville, agree to share the cost of a park that will be bordered
by the three cities. Because their populations are different, they agree that Bayside will pay 1¼ as
much as Avalon and Clarkville will pay half as much as Bayside pays. Solve this problem without
using algebra. Instead work with diagrams and/or quantitative reasoning.
a) What fractional part of the whole cost will each city pay?
b) What is the ratio of Avalon’s share to Clarkville’s share?
11. Re-write each of the given forms in the specified form.
a) Given: Quantity P is 3½ times as large as Quantity R.
Re-write in the ratio form P:R =
b) Given: M:N = 7:12
Re-write as Quantity N is ____ times a s much as Quantity M.
c) Given: Quantity N is 60% more than Quantity P
Re-write in the ratio form N:P =
12. a) Written as a decimal, 0.625% = ________ ; as a fraction, 0.625% = ________.
b) Written as a percent, 6.25 = ________ ; as a fraction, 6.25 = ________.
c) Written as a decimal, 6.25% = ________; as a fraction, 6.25% = ________ .
13. Two brothers, Larry and Moe, were given some candy by their older brother, Curly. Larry and Moe
are upset. Larry says, “I only got ¾ as much candy as you did, Curly”. Moe says, “Me too!” What
part of all the candy did Curly get?
14. In the year 2000, Smalltown’s budget was $100 million. In 2001, the budget figure was 20% higher
than the year 2000 budget and the 2002 budget was 10% higher than the 2001 budget. What was
the 2002 budget amount in millions of dollars?
15. One recipe to serve 8 people calls for 3 cups of rice. Since Al expects only 6 people to come for
dinner, he decides to use 1 cup of rice.
a) Explain Al’s thinking.
b) Give another way to decide how much rice is needed for 6 people.
c) Which way is better, your way or Al’s way? Why?
16. a) Is 7:4 the same as 6:3? Why or why not?
b) Is 4:3 the same as 8:6? Why or why not?
17. The Math Department uses 3 tablespoons of coffee with 5 cups of water to make coffee. The
Science department uses 5 tablespoons of coffee with 9 cups of water to make coffee. Which coffee
is stronger? Explain how you decide.
18. The big dog weighs 5 times as much as the little dog. The little dog weighs 3/8 as much as the
medium size dog. The medium sized dog weighs 10 pounds more than the little dog. How much
does the big dog weigh?
2
19. Adam and Matt are using different city road maps. On Adam’s map, a line 6 inches long represents
a road that is really 9 miles long. On Matt’s map, a line 8 inches long represents a distance that is
really 12 miles long. If both boys were to measure the distance from City A to City B along a line,
which would have the longer line in terms of inches? Explain how you decide.
20. Two workers spent 8 hours making a total of 216 parts. Worker A makes 13 parts in one hour. If
the workers work at a steady rate throughout the day, who is more productive - Worker A or Worker
B?
21. Solve each proportion. Which, if any, can be solved easily without using the cross multiplication
property?
5 13
4 n
4
n
n
7
8 24
=
b)
c)
d)
=
e) =
a)
=
=
10 n
5 3
7 21
n 15
10 9
22. Use a table to solve the problem: A candy store sells 4 pounds of chocolate for $5.60. How much
would 6 ¾ pounds of chocolate cost?
23. If the box shown below represents 75% of an amount, show a box that represents 125% of the same
amount and one that represents 50% of the same amount.
24. The decrease in the price of notebooks was $1.05 which was a 30% discount. What were the
original price and the new price?
25. The population of City A is 15,000 while the population of City B is 20,000.
a) The population of City A is what percent of the population of City B?
b) The population of City B is what percent of the population of City A?
c) The population of City A is what percent smaller than the population of City B?
d) The population of City B is what percent larger than the population of City A?
26. The police chief said, “Accidents are down in our town, with about 80% as many this year as last
year.” This year there were 725 accidents. If the police chief was correct, how many accidents were
there last year?
27. Show how to estimate each of the following and, if possible, predict (without actually calculating)
whether your estimate will be larger or smaller than the actual value:
a) 49% of 125 people
b) 15% of $39.15
c) 76% of 399 miles
d) 32 is 51% of what?
e) 21 is what % of 79?
3
28. If the platter of cookies shown below contains 3/10 of a box of cookies, how many cookies were in
the box?
29. a)
b)
Write an equation that states that 16 is a factor of the number k.
Write an equation that states that m is a factor of n.
30. Explain why each of the following is a composite number: 15, 63, 98 .
31. Complete the following addition and multiplication tables for even and odd numbers.
+
even
odd
a)
b)
c)
d)
even
odd
x
even
odd
even
odd
Can you make any definite statements about:
The sum of any number of even numbers?
The sum of any number of odd numbers?
The product of any number of even numbers?
The product of any number of odd numbers?
32. If 35 is factor on a number n, give two other factors of n besides 1 and n.
33. How many factors does each of the following have?
a) 34
b) 26 i32 i5
c) 2i5i7 i11
d) 2520
34. Consider the number m = 28 i133 i317 . Which of the following could not be factors of the number m.
Explain how you know.
a) 27 i11
b) 28 i132
35. The number 57,729,364,596 has too many digits for most calculators to display. Use divisibility tests
to decide which of the numbers 2, 3, 4, 5, 6, 8, 9, 10 and 11 divide this number.
36. Create a 6-digit number such that
a) 2 and 3 are factors of the number but 9 is not.
b) 2 is a factor of the number but 4 is not
c) 8 and 9 are factors of the number
37. Given the number 527,4 2, use divisibility tests to find all possible numbers that, when placed in
the
, make the number divisible by each of the following numbers:
a) 2
b) 3
c) 4
d) 5
e) 6
f) 8
g) 9
h) 10 i) 11
4
39. What, if anything, can you say about the oddness or evenness of m if
a) 5,063,338 x m is an even number
b) 5,063,338 + m is an even number
40. If n = 43759462138999999249 + 764321572, then is n an even number, or an odd number? Explain
your answer.
41. Circle T if the statement is true, F otherwise.
a) T F Every whole number is a multiple of itself.
b) T F It is possible for an even number to have an odd factor.
c) T F Zero is a multiple of every whole number.
d) T F 250 is a factor of 10030
42. a)
b)
T F There are no values of b and c for which 27b = 9c.
T F There are no values of r and s for which 11r = 9s.
43. Is there a whole number M that would make this true? If so, identify the number M, If not, explain
why not.
a) 35 i52 i173 = 34 i17 4 i M
b) 24 i72 i118 i22 = 25 i7 i116 iM
44. If a = 22 i34 i113 and b = 23 i3i56 i72 find LCM(a,b) and GCF(a,b). Express your answers in factored
form.
45. Use factor lists the set intersection method to find LCM(35,56 ) and GCF( 35,56 ) .
46. Use prime factorization to find LCM(35,56 ) and GCF( 35,56 ) .
47. State a divisibility test for 4 and explain why it works.
48. State a divisibilty test for 3 and explain why it works.
ANSWERS
1. a) 5:1
b) 5
c)
1
5
d)
5
6
e)
1
6
2.
a) Mom is 30 years older than her daughter. b) Mom is 2 21 times as old as her daughter.
3.
If the ratio of museums to parks is 0, there are no museums in town b/c 0 = 0n where n is any nonzero number representing the number of parks. This ratio can be 0 if and only if the numerator is 0.
paid
Job #1 pays $500 for 40 hours which means $12.50 per hours since Amount
= 40$500
= 12.50 .
# hours
hours
Likewise, Job #2 pays $450 per hour or $12.50 per hour. Pay rates are the same.
4.
5.
a) part-whole diagram model
Meg pays $12 for ¾ of a pound of lobster. Let the rectangle represent 1
pound of lobster. Subdivide into 4 parts. The first 3 parts (¾ of a pound
of lobster) cost $12, so each ¼ pound of lobster costs $4. This means
that the full pound costs 4 x $4 or $16.
$12 for ¾ pound
5
b) proportion model – here’s one possible set-up
Amount of lobster
price paid
>
3
4
$12
=
1
x
and solve to get x = $16
6.
a) Since Abe painted only 3/5 as much floor as George did,
Abe painted 3 parts for every 5 parts painted by George.
b) 53
c) 38
d) 53 e) $187.50 to Abe, $312.50 to George
7.
Table with 3 platters for 10 people allows
3
10
a) Section A is
b) Section B is
c) Section A is
9.
3
7
21
70
and
2
7
=
20
70
, the table with 3 platters for 10
of the whole sandwich.
1 31 times as long as Section
3
as long as Section B.
4
A.
It’s important to note that this problem involves both additive
and multiplicative comparisons. Kai’s statement the “Today
I’m 5 years older you (Mai) are” is an additive comparison.
She will always be 5 years older than Mai. Mai’s statement
that four years ago Kai was 50% older is a multiplicative
comparison, saying that at that time Kai was half again as
old as Mai or Kai was 1½ times as old as Mai. Since we
know the difference in their ages is always 5 years we can
determine that 50% of Mai’s age four years ago was 5 years
which means her age then was 10 years, making Kai 15
years old. Today, the girls are 4 years older making Mai 14
and Kai 19. Six years from now, Mai will be 20 and Kai will
be 25 so that the ration of Mai’s age to Kai’s age then will be
25:20 or 5:4.
10. Bayside pays 1¼ times as much as Avalon.
Clarkville pays half as much as Bayside.
Cutting each of Avalon’s and Bayside’s
payments in two and dividing Clarkville’s
payment into five provides a
“compatible” way to measure the
payments. There are 23 equal parts!
a)
b)
George’s
part
platter per person while table with 4 platters for 14
4
3
= 72 platter per person. Since 10
=
people allows 14
people provides just a little more food per person.
8.
Abe’s
part
Ages 4 years ago
Mai
5 yrs
Kai
Avalon
Bayside
Clarkville
Avalon
Bayside
Clarkville
8
10
Avalon pays 23
of the cost while Bayside pays 23
and Clarkville pays
The ratio of Avalon’s share to Clarkville’s share is 8:5.
11. a) P:R = 3 ½ : 1 or P:R = 7:2
b) 12/7
50% of
Mai’s
age 4
yrs ago
c) N:P = 8:5
6
5
23
.
12. a) 0.0625% = 0.00625; 0.0625% =
b) 6.25 = 625%; 6.25 =
c)
625
100000
or
1
160
25
4
6.25% = 0.0625; 6.25% =
6.25
100
or
1
16
13. Larry and Moe each received ¾ as much candy as Curly. This means that for each 4 candies Curly
4
got, Larry and moe each received 3 pieces. It looks like this and Curly gets 10
or 52 of the candy.
Curly
xxxx
Larry
xxx
Moe
xxx
14. Year 2000 budget was $100 million. Year 2001 budget was 20% nigher than year 2000 budget so it
was 120% of year 2000 budget or $120 million. Year 2002 budget was 10% higher than year 2001
budget so year 2002 was 110% of year 2001 budget or $132 million.
15. a) Al though if there were 2 fewer people he needed 2 fewer cups of rice.
b) Another way to decide is to use a proportion comparing number of cups of rice to number of
people being served. Use 38 = 6x where x = the number of cups needed for 6 people and
x = 2 ¼ cups
c) Proportional reasoning works better. If 8 peope need 3 cups of rice, 4 people would need half as
much so Al’s recommendation of 1 cup for 6 people makes no sense.
16. a) Although 7 is 3 greater than 4 and 6 is 3 greater than 3, the rations 7:4 and 6:3 are not equal
because 6:3 means for every 6 is one set you have 3 in another set as illustrated below. For each 6
stars, there are 3 circles. You could also show this as 2 stars for every 1 circle. If you had 7 stars,
you would need an extra ½ circle, not an extra whole circle.
▲▲▲▲▲▲
♥ ♥ ♥
▲▲▲▲▲▲
♥
♥
♥
b) 4:3 is the same as 8:6. You can use a diagram argument similar to the one above.
17. Math Department uses 3 Tbsp coffee with 5 cups of water or 3/5 Tbsp coffee per 1 cup water.
Science Department uses 5 Tbsp coffee with 9 cups water or 5/9 Tbsp coffee per 1 cup water. Both
rates are slightly larger than ½ Tbsp coffee per 1 cup water. For easy comparison, you can rewrite
both fractions with the common denominator 45. 3/5 = 27/45 while 5/9 = 25/45. The Math
Department makes the stronger coffee. Another way to compar is to note the 3/5 = 6/10 which is
1/10 bigger than ½ while 5/9 = 10/18 which is 1/18 bigger than ½. Again we see that the Math
Department coffee is stronger b/c 1/10 >1/18.
18. Big dog weights 5 times as much as little dog.
Little dog weighs 3/8 as much as medium dog.
Medium dog weights 10 pounds more that little dog.
Big dog weighs 30 pounds.
7
DONE IN CLASS!
19. Adam’s map use 6 inches to represent 9 miles so that each inch on Adam’s map covers 1½ miles.
On Matt’s map, 8 inches represents 12 miles so that Matt’s map also uses the scale 1 inch per 1½
miles. If both boys measured the distance from City A to City B along a line, they would both have
the same measurement in inches.
20. Worker B is more productive. She makes 14 parts per hour.
21. a) n =
12
5
b) n = 12
c) n = 26
d) n =
70
9
e) n = 5
22. 4 pounds of candy costs $5.60 . Find cost of 6 ¾ pounds using a table.
# pounds
4
2
1
½
¼
6¾
cost
$5.60
$2.80
$1.40
$0.70
$0.35
$10.85
comments
½ the cost of 4 pounds
½ the cost of 2 pounds
½ the cost of 1 pound
½ the cost of ½ pound
Total pounds & cost
23. Box shown here represents 75% of an amount.
Subdivide into three parts, each 25%
Use this information to make boxes showing
125% of the same amount >>>>>
50% of the same amount >>>>>>.
24. Original price was $3.50; new price is $2.45 .
25. a) 75%
b) 133 31 %
c) 25%
d) 33 31 %
26. Accidents were about 80% as many this year as last year and there were 725 accidents this year.
725 accidents this year = 80% of last year’s total :
725 ÷ 4 = 181.25 so about
Accidents this year:
181.25 x 5 ≈ 906 last year
Accidents last year:
8
27. a) 49% of 125 people is about ½ of 125 people = 62.5 but since we’re talking people and 49% < ½,
we’ll use 62. Since 1% of 125 people is 1.25 the actual value of 49% of 125 will be 1.25 less than
62.5 so our estimate of 62 will be larger than the calculated value of 49% of 125
b) 15% of $39.15 is about 10% of $39 plus 5% of $39. So that $3.90 + $1.95 = $5.85. This
estimate would be slightly smaller than the true value b/c we dropped the 15 cents in the
estimate.
Or you could estimate that $39.15 is a little smaller that $40 so 15% of $39.15 would be a little
less than 15% of $40 which is $4 + $2 = $6. This estimate will be larger than the actual value.
c) 76% of 399 miles will be close to 75% or ¾ of 400 which is 300. Although you wouldn’t do this
with young children, for our own understanding we can show that the estimate of 75% of 400 is
actually less than the actual value of 76% of 399. Here’s the argument, using the distributive
property. It shows that the estimate is 3.24 less than the actual value. Notice that we did not
need to calculate 76% of 399 to show this conclusion!
75% of 400 = (0.75) ⋅ ( 400 )
= ( 0.76 − 0.01 ) ⋅ (399 + 1 )
= ( 0.76 ) ⋅ (399 ) + ( 0.76 ) ⋅ (1 ) − ( 0.01 ) ⋅ (399 ) − ( 0.01 ) ⋅ (1 )
= ( 0.76 ) ⋅ (399 ) +
0.76
−
3.99
−
0.01
= ( 0.76 ) ⋅ (399 ) − 3.24
d) We can estimate the answer to the question “32 is 51% of what?” by noting that 32 is 50% of 64.
Our estimated answer of 64 will be a little large b/c we’re looking for a number for which 32 is a
little more than half the number. This means that half the desired number must be less than 32
so that the desired number is 2 times a value less than 32 and thus must be less than 64.
e) We can estimate the answer to “21 is what % of 79?” by thinking about the question “20 is what
% of 80?” We know the answer to this simpler question is 25%. It’s not difficult to see that 21
is actually 25% of 84 since 21 x 4 = 84. Since we’re looking at 21 as a percent of 79, which is
21
21
> 84
since each
smaller than 84, we know from our work with the relative size of fractions that 79
fraction involves the same number of parts (numerator = 21) but the sizes of the parts differ
1
1
> 84
since the unit fraction with the smaller denominator is the larger unit fraction).
( 79
28. NOTE: There was a cookie missing from the platter! The platter contains 21 cookies which amounts
3
of the box. This means that 101 of the box would be 31 of 21 cookies or 7 cookies. The whole
to 10
box would contain 10 times this amount or 70 cookies.
29. a) 16 ⋅ p = k where p is some whole number. b) m ⋅ p = n where p is some whole number.
30. For some reason, there was no problem 30 on the sheet. To avoid confusion, I’ll wait to renumber
until next semester.
31. 15 is a composite number because it has factors of 3 and 5 as well as 1 and 15.
63 is a composite number b/c in addition to factors of 63 and 1 it’s easy to see that 3 is a factor.
98 is a composite number b/c in addition to factors of 98 and 1 it’s easy to see that 2 is a factor.
9
32. Completed tables are shown below:
+ even
odd
even EVEN ODD
odd ODD
EVEN
x even
odd
even EVEN EVEN
odd EVEN ODD
a) The sum of any number of even numbers must be even b/c the sum of two even numbers is even
which means that each time another even number is added to the sum the result will be an even
number..
b) The sum of any number of odd numbers may be even or odd depending on the quantity of
numbers added. If an EVEN number of odd numbers is added, the sum will be even since the odd
numbers can be paired. Each odd number will have a “partner” and each pair of odd numbers added
produces an even result. If an ODD number of odd numbers is added, the sum will be odd since all
but one of the odd numbers can be paired. The sum of the “paired” odd numbers will be even but the
adding the final remaining odd number gives an odd result.
c) The product of any number of even numbers must be even b/c if even one of the multipliers is
even the product will automatically have a factor of 2 which makes the product even.
d) The product of any number of odd numbers must be odd b/c the product of the first two odd
factors is odd. Likewise, every other factor is odd so the continued multiplation produces an odd
results. There must be at least one even factor to make the product even.
33. If 35 is a factor of a number n, so are 5 and 7 since they are factors of 35.
34. a) 34 has five factors: 30, 31, 32, 33, 34. Using the rule in the text, since the highest power of 3 that
appears is the 4th power and 3 is the only prime factor, the total number of factors is 4 + 1 = 5
b) 26 i32 i5 has 42 factors.
26 is a factor of the number, so each of 7 powers 20, 21, 22, 23, 24, 25, 26 is also a factor.
32 is a factor of the number, so each of 3 powers 30, 31, 32 is also a factor.
51 is a factor of the number, so each of 2 powers 50, 51 is also a factor.
Any of the 3 powers of 3 can be matched withany of the 2 powers of 5 to create 3 x 2 = 6 factors
involving only powers of 3 and 5: 30·50 = 1, 30·51 = 5, 31·50 = 3, 31·51 = 15, 32·50 = 9, 32·52 = 225.
Now each of these 6 factors can be matched with the 7 powers of 2 to create 6 x 7 = 42 factors.
A list is included below so you can visualize.
20
21
22
23
24
25
26
30·50
20·30·50
21·30·50
22·30·50
23·30·50
24·30·50
25·30·50
26·30·50
30·51
20·30·51
21·30·51
22·30·51
23·30·51
24·30·51
25·30·51
26·30·51
31·50
20·31·50
21·31·50
22·31·50
23·31·50
24·31·50
25·31·50
26·31·50
31·51
20·31·51
21·31·51
22·31·51
23·31·51
24·31·51
25·31·51
26·31·51
32·50
20·32·50
21·32·50
22·32·50
23·32·50
24·32·50
25·32·50
26·32·50
32·51
20·32·51
21·32·51
22·32·51
23·32·51
24·32·51
25·32·51
26·32·51
c) 2i5i7 i11 has 2 x2 x 2 x 2 = 16 factors.
d) 2520 = 23·32·51·71 and has (3 + 1)(2 + 1)(1 + 1)(1 + 1) = (4)(3)(2)(2) = 48 factors.
10
35. a) 27 i11 cannot be a factor of m = 28 i133 i317 b/c 11 is not a factor of m
b) 28 i132 is definitely a factor of m = 28 i133 i317 b/c 28 i133 = 28 i132 i13 is a factor
36. Use divisibility tests to investigate which of the numbers 2, 3, 4, 5, 6, 8, 9, 10, and 11 are factors of
57,729,364,583. (I made a typo on this problem. Sorry!!!)
2 is NOT a factor b/c the number is not even. Note that if 2 is not a factor, neither is any other
even number. This means 4, 6, 8 and 10 are NOT factors of 57,729,364,583.
The sum of the digits of this number is 56 which is not divisible by either 3 or 9 so 3 and 9 are
also NOT factors of 57,729,364,583.
5 is not a factor b/c the number does not end in a 5 or a 0.
Use divisibility tests to investigate which of the numbers 2, 3, 4, 5, 6, 8, 9, 10, and 11 are factors of
57,729,364,572.
2 is a factor b/c the number is even
3 is a factor b/c the sum of the digits is 57and 3 is a factor of 57.
4 is a factor b/c 4 is a factor of the number formed by the last two digits, 72.
5 is not a factor b/c the number doen’t end in a 5 or in a 0.
6 is a factor b/c both 2 and 3 are factors.
8 is not a factor b/c 8 is not a factor of the number formed by the last three digits, 572.
9 is not a factor b/c the sum of the digits is 57 and 9 is not a factor of 57.
10 is not a factor b/c the number does not end in a 0.
37. Given the number 527,4 2, use divisibility tests to find all possible numbers that, when placed in
the
, make the number divisible by each of the following numbers:
a) 2
b) 3
c) 4
d) 5
e) 6
f) 8
g) 9
h) 10 i) 11
a) Since this number ends in 2 (which means it must be an even number), any of the numbers
0,1,2,3,4,5,6,7,8,9 can be placed in the box and the number is still divisible by 2.
b) The sum of the digits here is 5+2+7+4+
+2 = 20+
. Filling the box with 1, 4 or 7 makes the
sum of the digits, respectively, 21, 24 or 27, all of which are divisible by 3.
c) The number formed by the last two digits is
2. Filling the box with 1, 3, 5, 7 or 9 makes the last
two digits a number divisible by 4 and consequently makes the given number divisible by 4.
d) No matter what you put in the box, this number can’t be divisible by 5 b/c it doesn’t end in 5 or 0.
e) The number must be divisble by both 2 and 3 in order to be divisible by 6. All digits work for
divisibility by 2 but only 1, 4, and 7 give divisibility by 3. The three numbers also make the given
number divisible by 6.
f) The number formed by the last three digits is 4 2. Filling the box with 3 or 7 makes the
last three digits a number divisible by 8 and consequently makes the given number divisible by 8.
g) The sum of the digits here is 5+2+7+4+
+2 = 20+
. Filling the box with 7 makes the
sum of the digits 27, all of which are divisible by 9.
h) No matter what you put in the box, this number can’t be divisible by 5 b/c it doesn’t end in 0.
39. a) If 5,063,338 x m is an even number, m may be either even or odd. We know that 2 is a factor of
5.063,338, so 2 is also a factor of any product involving this number.
b) If 5,063,338 + m is an even number, m must be even. If m were odd, the sum would also be odd.
40. If n = 43759462138999999249 + 764321572, n must be an even number. Notice that 76432157 is
an odd number, so that squaring it gives (odd)(odd) = odd. Adding this odd square to the first
addedn which is also odd gives (odd) + (odd) = even for the number n.
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41. a) True
b) True
c) True
d) True
42. a) False. Since 27b = (33)b = 33b while 9c = (32)c = 32c, it follows that 27b = 9c if 3b = 2c or b = 23 c .
Some possible values for b and c are b=6, c=9 or b=12, c=18 or b=18, c=27. etc.
b) True, since 11 and 9 have no common factors
43. a) There is no number M that will make 35 i52 i173 = 34 i17 4 iM . If M included factors 3i52 the powers
of 3 and 5 would match. However, the rightside of the expression includes a factor of 17 4 while the
left side includes only 173 and we have no way to include something in M (which must be a whole
number) that will decrease the power of 17 on the right side.
b) The statement 24 i72 i118 i22 = 25 i7 i116 iM will be true if M = 7 i113 since
24 i72 i118 i22 = 24 i7 i7 i116 i112 i2i11
(
= 25 i7 i116 i 7 i113
)
44. If a = 22 i34 i113 and b = 23 i3i56 i72 , then LCM(a,b) = 23 i34 i56 i72 i113
and GCF(a,b) = 22 i3 .
45. Use factor lists the set intersection method to find LCM(35,56 ) and GCF( 35,56 )
Set of multiples of 35 = {35, 70, 103, 140, 175, 210, 245, 280, 315, …}
Set of multiples of 56 = {56, 112, 168, 224, 280, 336, 392, …}
So LCM(35, 56) = 280
Set of factors of 35 = {1, 5, 7, 35}
Set of factors of 56 = {1, 2, 4, 7, 8, 14, 28, 56}
So GCF(35,56) = 7
35 = 5i7 and 56 = 23 i7
46. Use prime factorization to find LCM(35,56 ) and GCF( 35,56 ).
so LCM(35,56) = 23 i5i7 = 280 You need to use the greatest power of all prime factors involved
and GCF(35,56) = 7. The only common factor is 7 and it appears once only in each number.
47. To check for divisibilty by 4, consider the number formed by the last two digits of the given number. If
this “last two digits” number is divisible by 4, so is the number itself. This works b/c we can rewrite
any number of three or more digits as the sum of the hundreds + the tens + the units. Since 100 is
divisible by 4, any multiple of 100 is also divisible by 4. Consequently, to check if the complete
number is divisible by 4, we need only check if the number that is formed in the tens and the units
place is divisible by 4.
Ex: 5246 = 5200 + 46. 5200 = 52(100) is divisible by 4 but 46 isn’t, so 5246 is not divisible by 4
Ex: 9728 = 9700 + 28. 9700 = 97(100) is divisible by 4 and so is 28, so 9728 is divisible by 4.
48. If the sum of the digits of a number is divisible 3, the number itself is divisible by three. Our place
value number system use powers of 10 and conveniently, each power of 10 is one larger than a
multiple of 9 which makes this rule work. Suppose ABCD represents a base ten number. In
expanded form, we would write
ABCD = (1000 )A + (100 )B + (10 )C + (1 )D
= ( 999 + 1 )A + ( 99 + 1 )B + ( 9 + 1 )C + (1 )D
= ( 999 )A + ( 99 )B + ( 9 )C + (1 )A + (1)B + (1 )C + (1 )D
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Rewriting each power of 10 as a multiple of 9 plus 1 and
using the distributive property, we see that since 3 divides
9 it also divides 999A, 99B and 99C. If 3 divides
A+B+C+D, then the entire number is divisible by 3!
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