Final Exam questions
On your final exam you will be given four questions: two from each part (two
theorems and two problems). You will be asked to solve any three out of four.
You should write your answers on a paper and present afterwards one by one.
Good luck!
First part. Prove the following:
Proposition 1.1. Let a, b, c  Z. If a | b and b | c , then a | c .
Proposition 1.2. Let a, b, c , m, n  Z. If c | a and c | b , then c | ma  nb .
Theorem 1.4. Let a, b  Z with b  0 . Then there exist unique q, r  Z such that
a  bq  r ,
0r b.
Lemma 1.5. Every integer greater than 1 has a prime divisor.
Theorem 1.6. (Euclid) There are infinitely many prime numbers.
Proposition 1.8. For any positive integer n , there are at least n consecutive
composite positive integers.
Proposition 1.10. Let a, b  Z with (a, b )  d . Then (a / d , b / d )  1 .
Lemma 1.14. (Euclid) Let a, b , p  Z with p prime. If p | ab , then p | a or p | b .
State the Fundamental Theorem of Arithmetic (Theorem 1.16).
State the Proposition 1.17 on how to calculate the gcd and lcm of two numbers.
Lemma 1.18. Let x, y  R . Then max{ x, y}  min{ x, y}  x  y .
Theorem 1.19. . Let a, b  Z with a, b  0 . Then (a, b) [a, b]  ab .
Proposition 2.4. Let a, b, c, d  Z such that a  b mod m and c  d mod m
(a) (a  c)  (b  d ) mod m
(b) ac  bd mod m
Proposition 2.5. Let a, b, c  Z . Then ca  cb mod m if and only if a  b mod (m /( c, m)) .
State the Chinese remainder Theorem (Theorem 2.9).
Lemma 2.10. Let p be a prime number and let a  Z . Then a is its own inverse
modulo p if and only if a  1 mod p .
Theorem 2.11 (Wilson’s theorem). Let p be a prime number. Then
( p  1)!  1 mod p .
Proposition 2.12. Let n  Z with n  1. If (n  1)!  1 mod n , then n is a prime
number.
Theorem 2.13 (Fermat’s little theorem). Let p be a prime number and let a  Z . If
p does not divide a , then
a p 1  1 mod p .
Part two. Solve the following problems.
Number Theory Problems.
1. Find integers a, b, and c such that a | bc but a does not divide b and a does not
divide c .
2. Find the unique integers q and r guaranteed by the division algorithm with
each dividend and divisor below.
(a) a  47, b  6 .
(b) a  281, b  13
(c) a  16, b  0
(d) a  105, b  10
3. If a, b  Z , find a necessary and sufficient condition that a | b and b | a .
4. Prove or disprove the following statements.
(a) If a, b, c and d are integers such that a | b and c | d , then a  c | b  d .
(b) If a, b, c and d are integers such that a | b and c | d , then ac | bd .
(c) If a, b and c are integers such that a does not divide b and b does not
divide c , then a does not divide c .
5. (a) Let a, b, c  Z with c  0 . Prove that a | b if and only if ac | bc .
(b) Provide a counterexample to show why the statement of part (a) does not
hold if c  0 .
6. Let a, b  Z with a | b . Prove that a n | b n for every positive integer n .
7. Find 13 consecutive composite positive integers.
8. Prove or disprove the following
Conjecture: There are infinitely many prime numbers p for which p  2 and p  4
are also prime numbers.
9. Find the greatest common divisors below.
(a) (18,36,63)
(b) (30,42,70)
(c) (0,51,0)
(d) (35,55,77)
10. Let a  Z with a  0 . Find the greatest common divisors below.
(a) (a, a n ) where n is a positive integer
(b) (a, a  1)
(c) (a, a  2)
(d) (3a  5, 7a  12)
11. Find four integers that are relatively prime (when taken together) but such that
no two of the integers are relatively prime when taken separately.
12. (a) Do there exist integers x and y such that x  y  100 and ( x, y )  8 ? Why or
why not?
(b) Prove that there exist infinitely many pairs of integers x and y such that
x  y  87 and ( x, y )  3 .
13. Let a, b  Z with a and b not both zero and let c be a nonzero integer. Prove that
(ca, cb)  c (a, b) .
14. Let a, b  Z with (a, 4)  2 and (b, 4)  2 . Find (a  b, 4) and prove that your answer
is correct.
15. Let a, b, c  Z with (a, b)  1 and c | a  b . Prove that (a, c)  1 and (b, c)  1 .
16. Use the Euclidean algorithm to find the GCD
(a) (37, 60)
(b) (78, 708)
(c) (441, 1155)
17. Find two rational numbers with denominators 11 and 13, respectively, and a
sum of
7
.
143
18. Find the GCD and the LCM of each pair of integers below
(a) 2 2  33  5  7, 2 2  3 2  5  7 2
(b) 2 2  5 7  1113 , 3 2  7 5  1311
19. Find all pairs of positive integers a and b such that (a, b)  12 and [a, b]  360 .
20. Find all positive integers m for which the following statements are true.
(a) 13  5 mod m
(b)  7  6 mod m
21. Let a, b  Z such that a  b mod m . If n is a positive integer such that n m , prove
that a  b mod n
22. Find all least nonnegative incongruent solutions of the following congruences.
(a) 9 x  21 mod 30
(b) 18 x  12 mod 28
23. Find all least nonnegative incongruent solutions of the following congruences.
(a) 18 x  15 mod 27
(b) 12 x  16 mod 32
24. Find the inverse modulo m od each integer n below.
(a) n=5, m=26
(b) n=51, m=99
25. Find the inverse modulo m of each integer n below.
(a) n=8, m=35
(b) n=40, m=81
26. Find the least nonnegative solution of each system of congruences below.
x  2 mod 5
(a) x  3 mod 7
x  1 mod 8
x  1 mod 7
(b) x  4 mod 6
x  3 mod 5