T101 Number Theory Review
1. Use the set intersection method to find the
following:
a)
GCD (8, 18) b) LCM (6, 8) c) GCD (16, 51)
d)
LCM (7, 9) e) GCD (18, 36, 54) f) LCM (15,
35, 42)
2. Find the prime factorization of the following
numbers:
a)
216 b) 2940 c)
825 d) 198,198
3. Determine whether the following are true or
false using divisibility tests.
Don't use a calculator!
a)
6 | 89 b) 15 | 10,000 c) 4 | 15,000 d) 12 | 32,304
e)
24 | 325,608 f)
45 | 13,075 g) 40 | 1,732,800 h)
36 | 677,916
4. Determine True or False:
a)
3 | 9 b) 12 | 6 c) 3 is a divisor
of 21 d) 6 is a factor of 3
e)
4 is a factor of 16 f) 0 | 5 g) 11 | 11 h) 48 is a multiple of 16
5. Test for divisibility by 11:
a) 2838 b)
71,992 c) 172,425
6. Use Eratosthenes Sieve to find all the
primes less than 150.
7. The customer said to the cashier: "I have 5 apples at 27 cents each and 2
pounds of potatoes at 78 cents per pound.
I also have 3 cantaloupes and 6 lemons, but I don't remember the price
for each." The cashier said,
"That will be $3.52." The
customer said, "You must have made a mistake." The cashier checked and the customer was
correct. How did the customer catch the
mistake?
8. Use any method to find the following:
a)
GCD (12, 60, 90) b) LCM (16, 8) c) GCD (55, 75,
245)
d)
LCM (2, 3, 5) e) GCD (15, 35, 42) f) LCM (20, 36, 42, 33)
9. Use the Euclidean algorithm to find GCD,
then also find LCM.
a)
24, 54 b) 39, 91 c) 72,
160
10. Test for divisibility by 2, 3, 4, 5, 6, 8,
9, 10, 11
a)
11,223,344 b) 6,543,210 c) 6944 d) 81,432
e)
1,076,770 f) 50,177 g)
1,161,914 h) 150,024
11. What is the smallest natural number
divisible by all the natural numbers 1 through 9?
12. Which of the following are primes? Justify your answers.
a)
149 b) 89 c)
87 d) 43
e) 737 f) 411 g)
91 h) 1003
13. What is the largest prime you need to use in
checking if 689 is a prime?
in testing 7001?
14. Find the smallest natural number that has
exactly the following number of distinct prime
(that is, different) divisors, and
describe the procedure used to determine the answer.
a)
two b) three c) four d) five
15. Find the smallest natural number that has
exactly the following number of distinct,
not necessarily prime, divisors.
a)
two b) three
c) four d) five e) six f)
seven
16. Two joggers are running around a one mile
track. One jogger does a mile in 4
minutes, but the second one takes 10 minutes.
If they start at the same time and same place, how long will it take
them to be at this place together if they continue to run?
ANSWERS
1. a)
Divisors of 8 = {1, 2, 4, 8} b) Mult of
6 = {6, 12, 18, 24, 30,...}
Divisors of 18 = {1, 2, 3, 6, 9,
18} Mult of 8 = {8, 16, 24, 32,
40,...}
Greatest Common Divisor = 2 Least
Common Multiple = 24
c)
1 d) 63
e) 18 f) 210
2. a) 
b) 
c) 
d) 
3. a)
False b) False
c) True d)
True e) True
f) False g)
True h) True
4. a)
True b) False
c) True d)
False e) True
f) False g)
True h) True
5. a)
True b) False
c) True
6. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
101, 103, 107, 109, 113, 127, 131, 137,
139, 149
7. Of all the 'price-quantity' items, either
the price or the quantity is a multiple of 3, but $3.52 is not.
8. a)
6 b) 16
c) 5 d) 30 e)
1 f) 13860
9. a)
GCD = 6 LCM = 216 b)
GCD = 13 LCM = 273 c)
GCD = 8 LCM =1440
10. a)
2, 4, 8, 11 b) 2, 3, 5, 6, 10 c) 2, 4, 8 d)
2, 3, 4, 6, 8, 9
e)
2, 5, 10 f) None g)
2 h) 2, 3, 4, 6, 8
11. 
12. a)
Prime b) Prime c) Composite 
d) Prime
e)
Composite 
f) Composite 
g)
Composite 
h) Composite 
13. 23;
83
14. a)
2 b) 
c) 
d) 
15. a) 
b) 
c) 
d) 
e) 
f) 
16. LCM(10, 4) = 20 minutes