Question

We can find a system of congruences where x≡a (mod 4) x≡b (mod 5) where the solution is x≡7 (mod 20).

Answer 1

If x ≡ 7 (mod 20), then

x ≡ 7 ≡ 27 ≡ 47 ≡ 67 ≡ 87 ≡ … (mod 20)

or equivalently, x = 7 + 20k for some integer k.

Taken mod 4, we have

x ≡ 7 + 20k ≡ 3 (mod 4)

and mod 5,

x ≡ 7 + 20k ≡ 2 (mod 5)

so that a = 3 and b = 2.