Prove by mathematical induction
\[
5^{2n} - 1
\]
is divisible by 24 for all natural numbers \( n \gt 0 \).
Prove the statement is true for a particular n. Here n = 1 is required.
To show this is true for n=1, observe that the expression = 24. Now 24 is divisible by 24, so the statement is true for n = 1.
Assume the statement is true for n = k and show it is true for n = k + 1.
To show the statement is true for n = k+1,
\[ \begin{align}
5^{2(k+1)} - 1 &= 5^{2k+2} - 1 \\
&= 5^{2k} \times 5^2 - 1 \\
\end{align} \ \]
Use the inductive hypothesis to simplify the right hand side.
By the inductive hypothesis,
\[ \begin{align}
5^{2k} \times 5^2 - 1 &= (24m + 1) \times 25 - 1 \quad \text{for some integer m}\\
&= 24 \times 25 m + 25 - 1 \\
&= 24(25m + 1)
\end{align} \ \]
Which is divisible by 24. So the statement is true for n = k+1.
By the principle of mathematical induction,
The statement is true for \( n \gt 0 \).
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