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Answers to the Final Review
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1. The statement “If 5 | 20, then 5 | 200” is an instance of what theorem? Divisibility
of a product theorem.
2. The statement “If 5 | 20 and 5 | 40, then 5 | 60” is an instance of what theorem?
Divisibility of a sum theorem.
3. According to the divisibility-of-a-difference theorem, if 7 | 49 and 7 | 91, then 7 |
(91 – 49), or 7 | 42.
4. What are the multiples of 7? The multiples of 7 are 0, 7, 14, 21, …(also –7, -14, 21, …) In general, 7n with n an integer.
5. Explain why 3 is not a multiple of 6. 3 is not a multiple of 6 because there is no
integer n such that 6n = 3.
6. Name a number that is a factor and a multiple of 15. 15
7. Use the prime number test to classify the following numbers as prime or
composite: 71, 91, 577, 697, 1,523. The primes are 71, 577, 1,523.
8. Find the prime factorization of the following numbers: 495, 320, 90, 3155.
495 = 32 μ 5 μ 11, 320 = 26 μ 5, 90 = 2 μ 32 μ 5, 3155 = 5 μ 631.
9. Which of the following pairs of numbers are relatively prime?
a. 11 and 12
b. 34 and 51
c. 157 and 46
Two numbers are relatively prime if GFC = 1. So, a) and c) are relatively prime.
10. Find the GCF for each of the following, using factor-tree method and the
Euclidean algorithm.
a. 627 and 665
b. 851 and 2035
c. 551 and 609
GCF(627, 665) = 19
GCF(851, 2035) = 37
GCF(551, 609) = 29
11. Find the LCM for each of the following:
a. 20 and 32
b. 14 and 20
c. 10 and 12
LCM(20, 32) = 160
LCM(14, 20) = 140
LCM(10, 12) = 60
12. Draw a number line showing the following fractions: 1/3, 2/3, 5/8, 7/8, 3/4, 2/9,
4/19, 7/15, 7/10.
Your number line should show the fractions in the following order:
4/19, 2/9, 1/3, 7/15, 5/8, 2/3, 7/10, 3/4,7/8
13. Compute the following without a calculator, and simplify:
a. 5 3/4 + 2 5/8
b. 3 5/12 + 7 2/3
c. 10 1/6 – 5 2/3
d. 90 1/3 – 32 7/9
5 3/4 + 2 5/8 = 23/4 + 21/8 = 46/8 + 21/8 = 67/8 = 8 3/8
3 5/12 + 7 2/3 = 41/12 + 23/3 = 41/12 + 92/12 = 133/12 = 11 1/12
10 1/6 – 5 2/3 = 61/6 – 17/3 = 61/6 – 34/6 = 27/6 = 4 3/6 = 4 1/2
90 1/3 – 32 7/9 = 271/3 – 295/9 = 813/9 – 295/9 = 518/9 = 57 5/9
14. Compute the following without a calculator, and simplify:
a. 5/8 π 1/2 = 5/8 μ 2 = 10/8 = 5/4
b. –8 π 3 1/9 = -8 π 28/9 = -8 μ 9/28 = -72/28 = -18/7 = -2 4/7
c. 3/5 π 7/10 = 3/5 μ 10/7 = 30/35 = 6/5 = 1 1/5
d. 10 π 4 2/3 = 10 π 14/3 = 10 μ 3/14 = 30/14 = 2 1/7
15. Compute the following without a calculator, and simplify:
a. 3/8 μ 1/6 = 3/48
b. 1/2 μ 4/3 = 4/6 = 2/3
c. 9/10 μ 3/2 = 27/20 = 1 7/20
d. 3 1/2 μ 4 1/8 = 7/2 μ 33/8 = 231 / 16 = 14 7/16
e. 2 1/6 μ 3 1/3 = 13/6 μ 10/3 = 130/18 = 7 4/18 = 7 2/9
16. Write the following fractions as decimals:
a. 3/5 = 0.6
b. 2 2/9 = 2.222….
c. 7/20 = 0.35
d. 7/11 = 0.63636363…
17. Write the following decimals as fractions:
a. 0.731 = 731/1000
b. –13.04 = -13 4/100 = -1304/100
c. –0.013 = - 13/1000
d. 4.712 = 4 712/1000 = 4712/1000
18. Write the following decimals as fractions:
a. 52.353535353535….. = 52 + 35/99 = 5183/99
b. 0.3917391739173917…. = 3917/9999
c. 0.999999….. = 1
d. 0.211111….. = 2/10 + 1/90 = 19/90
e. 0.3414141….. = 3/10 + 41/990 = 338/990
f. 0.534343434…. = 5/10 + 34/990 = 529/990
g. 0.1233333333…. = 12/100 + 333/99900 = 12321/99900
19. Classify the following numbers as rational or irrational:
a. 3/7
rational
b. –3
rational
c. ◊(4/9) rational
d. ◊7
irrational
e. 0.33333…………. rational
f. 0.817 rational
g. 0.4263827564…. irrational
h. 0.121121112…. irrational
20. What is the multiplicative inverse of 0.1? The multiplicative inverse of 0.1 is 10,
because 0.1 times 10 equals 1.