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Number theory involving multiples of 2023

Source: Irish Mathematical Olympiad 2023 Problem 5

May 16, 2023
number theory

Problem Statement

The positive integers a,b,c,da, b, c, d satisfy
(i) a+b+c+d=2023a + b + c + d = 2023 (ii) 2023 ∣ ab−cd2023 \text{ } | \text{ } ab - cd (iii) 2023 ∣ a2+b2+c2+d2.2023 \text{ } | \text{ } a^2 + b^2 + c^2 + d^2.
Assuming that each of the numbers a,b,c,da, b, c, d is divisible by 77, prove that each of the numbers a,b,c,da, b, c, d is divisible by 1717.
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