Let p and q be primes with q>p such that p is not a factor of q-1. As q is a prime, we know that for any integer x we can write x^(q-1) in the form x^(q-1) = kq+1 for some integer k. Furthermore, since the greatest common divisor of p and (q-1) is 1, we can write 1 = ap + b(q-1) for some integers a and b.
(a) Show that every integer x can be written in the form x = y^p + jq for some integer y and some integer j.
My approach : since I know that x^(q-1) = kq+1 then x is a primitive root mod q ie x^ϕ(q)≡1(mod q). Since p is a prime too, there must exist a primitive root mod p as well. Let's call it y. hence y^ϕ(p)≡1(mod p).
So from the previous equations i can get y^(p-1)=hp+1, x^(q-1)=kq+1 and I have given 1=ap+b(q-1). I thought about putting this into a system but didn't help.