IMO 1984 SL 12

Find two positive integers a, b such that none of the num-

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IMO 1984 SL 12

Origin: NET

Problem

Find two positive integers a, b such that none of the num- bers a, b, a + b is divisible by 7 and (a + b)7 −a7 −b7 is divisible by 77.

Solution

By the binomial formula we have (a + b)7 −a7 −b7 = 7ab[(a5 + b5) + 3ab(a3 + b3) + 5a2b2(a + b)] = 7ab(a + b)(a2 + ab + b2)2. Thus it will be enough to find a and b such that 7 ∤a, b and 73 | a2+ab+b2. Such numbers must satisfy (a + b)2 > a2 + ab + b2 \geq73 = 343, implying a + b \geq19. Trying a = 1 we easily find the example (a, b) = (1, 18).