gcd(m,n) and modular arithmetic

"If gcd(m,n) = 1, show that for any pair of integers a and b there exists an integer x such that x ≡ a mod m and x ≡ b mod n."

I'm having trouble interpreting this problem. When I look at this, I think, for example:

m = 7
n = 11
a = 6
b = 8

and

x ≡ 6 mod 7 -> 6
x ≡ 8 mod 11 -> 8

I'm not sure what I'm doing wrong. If someone could show me an example of integers m, n, a, b where the problem works I think I can finish the problem. Any response is appreciated.

Let mp+nq=1, then

amp+anq = a
bmp+bnq = b

So

anq = a (mod m)
bmp = b (mod n)