If n, m are two relatively prime positive integers and a,b are any integers, then prove that: There exists an integer X such that 0 ≤ X < nm and X ≡ a(modn) and X ≡ b(modm).
Hint: Start with two integers x,y such that xn + ym=1 (explain why they exist), and then consider the integer bxn + aym.