Assume toward a contradiction that x and y are integers satisfying 24x – 5y = 10,
11x – 9y = 13. Then reducing both equations modulo 7 gives 3x + 2y ≡ 3 mod 7,
4x + 5y ≡ 6 mod 7. Adding these two congruences gives 7x + 7y ≡ 9 mod 7, which is equivalent to 0x + 0y ≡ 2 mod 7. Hence 0 ≡ 2 mod 7, which is a contradiction.