Prove by induction:

Use the Euclidean algorithm to determine the g.c.d. of 432 and 831. Then reverse the calculations to write the g.c.d. as a linear combination of the two.

Show that the equation 3x² - y³ = 176 has no solutions with x and y integers, by considering the equation in \Bbb Z₉.

Show that if n is odd, then the g.c.d. of n and n+8 is always 1. (Hint: show that any k > 1 that divides n can't divide n+8.)

Show that a² º 16(mod 10) implies a² º 16(mod 20).

(Hint: show that 10|(a-4)(a+4) implies 5 divides one of the factors and 2 divides both of them ( a-4 is even if and only if a+4 is even!).)

Use the Euclidean algorithm to find d = (217,133) and find integers x, y such that d = 217x + 133y.

Find the least non-negative residue of 3¹¹⁶ (mod 29).

Let p be a prime integer and suppose for some a Î \Bbb Z_p that a² = a. Prove that a = [0]_p or a = [1]_p in \Bbb Z_p. Also, give an example to show that this can be false if p is not a prime.

Prove by mathematical induction that 3 is a divisor of 2^{2n + 1} + 1 for every positive integer n.

Prove that [Ö15] is irrational.

Find the smallest positive integer in the set {10u + 15v : u,v Î \Bbb Z}. Write a sentence or two justifying your answer.

Prove that if a,b and c are integers such that a|b and a|(b + c) then a|c.

What is the remainder when one divides (127)(244)(14)(-45) by 13? (You don't need to actually perform long division.)

If p is a positive prime number and p|a², prove that p|a. (Be sure to state completely any definition or theorem you use.)

Prove: If [a] = [1] in \Bbb Z_n, then (a,n) = 1.