djb86 19.04.2013 21:21 Let $m$ and $n$ be positive integers. Show that $25m+ 3n$ is divisible by $83$ if and only if so is $3m+ 7n$.
fractals 19.04.2013 21:23 Solution Multiply the first number by $30$, giving $750m + 90n$, then reduce mod $83$, giving $3m + 7n$. EDIT: sure
mrdriller 29.11.2020 08:52 Suppose that 83|25m+3n then 83|25m-80n then 83|5m-16n then 83|88m-16n then 83|11m-2n then 83|-72m-2n then 83|72m+2n then 83|36m+n then 83|36m+84n then 83|3m+7n \leftrightarrow then 83|3m+7n then 83|3m+83*9m+7n+83n then 83|750m+90m then 83|25m+3n.