Question

Show that 2^4n-1 is divisible by 15, if n be a positive integer

Answer 1

$2^{4n}=16^n=(15+1)^n$

From binomial theorem we have
$(15+1)^n=15M+1$ for some integer $M$

Thus,
$2^{4n}-1=15M+1-1=15M$

Answer 2

$N=2^{4n} - 1\ is\ divisible\ by\ 15,\ means\ N\ is\ divisible\ by\ 3\ and\ by\ 5.\\\\Proof\ by\ induction\\\\Let\ n=1,then\ N=2^{4*1}-1=15,\ divisible\ by\ 15\\\\let\ N\ be\ divisible\ by\ 15\ for\ n.\\\\N(n+1)=2^{4(n+1)}-1=2^{4n+4}-1=2^{4n}*2^4-1=16*2^{4n}-1\\.\ \ \ =16*2^{4n}-16+15\\.\ \ \ =16(2^{4n}-1)+15\\.\ \ \ =16\ N+15\\\\Since,\ N\ is\ divisible\ by\ 15,\ N(n+1) \ is\ also\ divisible\ by\ 15$