Proof that pairs of numbers whose GCF=1 can generate all subsequent integers

Question

Prove that any linear combination of pairs of relatively prime numbers (that is pairs of numbers whose GCF is 1) will, beyond a certain number, generate all subsequent integers.

*This does not hold true for other pairs of integers, i.e. pairs of integers with a GCF greater than 1.

Answer 1

If a and b are integers with the property that their gcd is 1 then we know that there are integers x, y such that xa+yb=1. But then, if k is any other integer we have that k=kxa+kyb which means that k can be written as a linear combination of the integers a, b which finishes the proof of the claim.