Prove: Use R - 3L for divisibility by 31.
N=10R+LR−3L=−3(10R+L)+31RThus 31∣R−3L⇔31∣3(10R+L)⇔31∣10R+L=N(we use the fact that 3and 31are mutually prime)N=10R+L\\R-3L=-3\left( 10R+L \right) +31R\\Thus\,\,31|R-3L\Leftrightarrow 31|3\left( 10R+L \right) \Leftrightarrow 31|10R+L=N\\\left( we\,\,use\,\,the\,\,fact\,\,that\,\,3and\,\,31are\,\,mutually\,\,prime \right)N=10R+LR−3L=−3(10R+L)+31RThus31∣R−3L⇔31∣3(10R+L)⇔31∣10R+L=N(weusethefactthat3and31aremutuallyprime)
