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existence, x+2y+3z+7t=a and y+2z+5t=b

Source: Yugoslav TST 1981 P3

May 29, 2021
Diophantine equationnumber theory

Problem Statement

Let a,ba,b be nonnegative integers. Prove that 5a>7b5a>7b if and only if there exist nonnegative integers x,y,z,tx,y,z,t such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}