Prove by the principle of mathematical induction that for all `n in N ,3^(2n)` when divided by `8` , the remainder is always `1.`

Explanation

Let P(n) be the statement given by

`P(n):3^(2n) `

`P(n):3^(2n)=8lambda+1 `

`P(1):3^2=8lambda+1=3^2=8xx1+1=8lambda+1, `

so,P(1) is true.

We shall now show that
`P(m+1) i.e.
3^(2(m+1))=8mu+1` for some `muinN`.

Now,
`3^(2(m+1))=3^(2m)×3^2=(8lambda+1)×9`

`=72lambda+9=72lambda+8+1=8(9lambda+1)+1=8mu+1,` where` mu=9lambda+1`

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