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

Question

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

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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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