Answer to Question #326610 in Combinatorics | Number Theory for Allen

Question

Prove: Use R + 7L to determine if 23| N.

Answer 1

$N=10R+L\\R+7L=7\left( 10R+L \right) -69R=7\left( 10R+L \right) -23\cdot 3R\\Thus\,\,23|R+7L\Leftrightarrow 23|7\left( 10R+L \right) \Leftrightarrow 23|10R+L=N\\\left( we\,\,use\,\,the\,\,fact\,\,that\,\,7and\,\,23are\,\,mutually\,\,prime \right)$