Saudi Arabia Mathematical Competitions

Question

Let a, b, c, d be positive integers such that a + b + c + d = 2011. Prove that 2011 is not a divisor of a b - c d.

Accepted Answer

We have (a + c)(b + c) = a b + a c + b c + c^2 = (a + b + c + d) c + a b - c d = 2011 c + a b - c d Because 2011 is a prime, if 2011 a b - c d, then 2011 a + c or 2011 b + c. This is not possible since 0 < a + c < 2011, and 0 < b + c < 2011, a contradiction.