How to prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m? m, t, k are Natural numbers.
How can I prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m ?

Hint $\rm\ mod\ 67\!:\ t\,\equiv\, 20\,k\:\Rightarrow\: \color{#C00}{10\,t}\,\equiv\, 200\,k\, \equiv\, (3\!\cdot\! 67\!-\!1)\,k\,\equiv\, \color{#C00}{-k}\:\Rightarrow\: \color{#C00}{10\,t\!+\!k}\,\equiv\, 0$
