Prove by induction that $7^{2n}-48n-1$ is divisible by 2304 $$P(n):2304\mid7^{2n}-48n-1$$
I've done the base case; $P(1)$ is true because the expression then evaluates to zero, which is divisible by 2304. Now I'm stuck on the inductive step: proving $P(m+1)$ true if $P(m)$ is true. I do know this though:
$$7^{2m+2}-48(m+1)-1
=49\cdot7^{2m}-48m-49$$

Hint $\ $ Conceptually, the induction is simply the first two terms of the Binomial Theorem, namely put $\ a = 48\ $ below and note $\,48^2 = 2304$
${\rm mod}\ a^2\!:\  (1\!+\!a)^n \equiv 1\! +\! na\ $ is true for $\ n = 0\ $ and the induction step is easy:
$\qquad\quad\ (1\!+\!a)^{n+1}\! = (1\!+\!a)(1\!+\!a)^n\overset{\rm induct}= (1\!+\!a)(1\!+\!na) \equiv 1\!+\!(n\!+\!1)a\ $ by $\ a^2\equiv 0$
