Question

Prove that $(106{)}^{85}-(85{)}^{106}$ is divided by 3 ?

Answer 1

$L.H.S.=(1+105{)}^{85}-(1+84{)}^{106}$
1,1 will cancel and both 105 and 84 are divisible by 3 and binomial coefficients which occur are all integers and hence L.H.S. is divisible by 3.