math 420 First Mid-term  NO CALCULATORS  Friday October 9, 1987

(15) 1. a) State Wilson’s Theorem.

b) Illustrate the idea of the proof of Wilson’s Theorem by using the example p = 13.

(20) 2. a) Find the smallest integer n such that n ≥ 8 and n ≡ 7 (mod 4),

n ≡ 7 (mod 5), and n ≡ 7 (mod 13).

b) Given that 12x ≡ 1 (mod 59) has x = 5 as a solution, solve each of the following:

i) 12x ≡ 8 (mod 59)
ii) 12x ≡ 13 (mod 59)
iii) 12x ≡ 35 (mod 59).

(15) 3. Let a and b be positive integers. Prove that there exist x and y with a = (x, y), b = [x, y] if and only if a|b.

(25) 4. a) Prove that the Diophantine equation ax + by = c has a solution if and only if

the congruence ax ≡ c (mod b) has a solution.

b) Let g = (a, b). By Theorem 1.4, ax + by = g has a solution.

Use this to prove that ax + by = c has a solution if and only if g|c. (Do not use theorems about congruences.)

(25) 5. a) Prove that for any n, n^{9} − n is divisible by 10.

b) Prove that n^{9} − n is divisible by 20 if and only if n does not have the form 4k + 2.