How do I prove this: Show that if x ≡ 1(mod n), then x^2 ≡ 1(mod n) and x^3 ≡ 1(mod n) and how to I explain this: explain why this argument should extend to show that if x ≡ 1(mod n), then x^k ≡ 1(mod n) for all k ∈ N

Let x, y ∈Z. Prove that if x≡1(mod 5) and y ≡2(mod5 ), then x^2-y^2≡0(mod 5).  Give 2 examples to show that this scenario either does or does not work. Give the proof with an explanation to each step.

All integers prove n> 1, and x = y (mod n), then x^2 = y^2 (mod n)

Prove that (50!)2 mod 101 = -1 mod 101.

Which are true for modulo 12? 1.) Does 5x ≡ 6 mod 12 have a solution? How many? 2.) Does 4x ≡ 6 mod 12 have a solution? How many? 3.) Does x^2+ 1 ≡ 0 mod 12 have a solution? How many? 4.) If xy ≡ 0 mod 12 does it follow that x ≡ 0 mod 12 or y ≡ 0 mod 12?

Find the least residue of 99! (Mod 101)

please do 8

Fand all Pntegers x such that Sx=!2 (mod チ) LOXEA (modst) 子« =16 (modlo)

please do 7

pl do not copy proof already here. It is wrong Let p and q be prime numbers such that q = 2p + 1, and p = 3 (mod 4). Prove that 2^p- 1 is a prime number (hence, a Mersenne prime!) if and only if p = 3. Hint. The "if" direction is easy. For the "only if" direction, proceed by proof by contradiction: by combining p= 3 and primality of 2^p - 1 derive that q must divide 2^p - 1.