Number & Algebra: Proof by Mathematical Induction

For $n = 1$: $5^{2(1)} - 1 = 5^2 - 1 = 25 - 1 = 24$ Since $24 = 24 \times 1$, the statement is true for $n = 1$.

Assume that $5^{2k} - 1$ is divisible by 24 for some integer $k \ge 1$. That is, $5^{2k} - 1 = 24M$ for some integer $M$, which implies: $5^{2k} = 24M + 1$

Consider the expression for $n = k + 1$: $5^{2(k+1)} - 1 = 5^{2k+2} - 1 = 5^{2k} \cdot 5^2 - 1 = 25(5^{2k}) - 1$ Substitute the inductive hypothesis $5^{2k} = 24M + 1$: $= 25(24M + 1) - 1 = 25(24M) + 25 - 1 = 24(25M) + 24 = 24(25M + 1)$ Since $M$ is an integer, $25M + 1$ is an integer. Thus, $5^{2(k+1)} - 1$ is divisible by 24.

Since the statement is true for $n = 1$, and if true for $n = k$ it is also true for $n = k + 1$, by the principle of mathematical induction, $5^{2n} - 1$ is divisible by 24 for all $n \in \mathbb{Z}^+$.