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

Suppose that a and b are integers, a ≡ 7 (mod 19), and b ≡ 5 (mod 19), Find the integer c with 0 ≤ c ≤ 18 such that: 12. (a − b) ≡ c mod 19 13. (7a + 3b) ≡ c mod 19 14. (2a2 + 3b2) ≡ c mod 19 15. (a3 + 4b3) ≡ c mod 19 please show your solutions :) I've already seen other answers to the problems but I'm unsure if that's correct pls help

Show that ()-{ = 1 -1 if q = ±1 mod 12 if q = ±5 mod 12

how would you prove this

Prove that if a + x ≡ a +y (mod n), then x ≡y (mod n)   details plz

Due in 15 minutes I need help!!

Find (177 mod 31 + 270 mod 31) mod 31. Show your work.

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.

Prove that there is no integer n such that n ≡ 4 (mod 10) and n ≡ 3(mod 15)

Find the least residue of 99! (Mod 101)