5^ 2n - 1 is divisible by 24 for all n . - Vidyadip

Question

$5^{2n} - 1$ is divisible by $24$ for all $\text{n}\in\text{N}.$

✓

Answer

Let $P(n)$ be the given statement.
Now,
$p(n): 5^{2n} - 1$ is divisible by $24$ for all $\text{n}\in\text{N}.$
Step 1:
$p(1): 5^2 - 1 = 25 - 1 = 24$
It is divisible by $24.$
Thus, $p(1)$ is true.
Step 2:
Let $P(m)$ be true.
Then, $5^{2m} - 1$ is divisible by $24.$
Now, let $5^{2\text{m}}-1=24\lambda,$ where $\lambda\in\text{N}.$
We need to show that $p(m + 1)$ is true whenever $p(m)$ is true.
Now,
$P(m + 1) = 5^{2m+2} - 1$
$= 5^{2m}5^2 - 1$
$=25(24\lambda+1)-1$
$=600\lambda+24$
$=24(25\lambda+1)$
It is divisible by $24.$
Thus, $p(m + 1)$ is true.
By the principle of mathematical induction, $p(n)$ is true for all $\text{n}\in\text{N}.$

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