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 3x^{2} - y^{3} = 176 has no solutions
with x and y integers, by considering the equation in \Bbb Z_{9}.

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 a^{2} º 16(mod 10) implies a^{2} 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 3^{116}
(mod 29).

Let p be a prime integer and suppose for some
a Î \Bbb Z_{p} that a^{2} = 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|a^{2},
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.

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