Prove by induction that $7^{2n}-48n-1$ is divisible by 2304

Question

$$P(n):2304\mid7^{2n}-48n-1$$ 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: $$7^{2m+2}-48(m+1)-1 =49\cdot7^{2m}-48m-49$$

Answer 1

$$7^{2(m+1)} - 48(m+1) - 1 = 49 \cdot 7^{2m} - 48m - 49 = 49\left(7^{2m} - 48m - 1\right) + 2304m$$

Answer 2

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 $\ a^2\equiv 0$