Hello, iceman1!

I can help with the first problem . . .

1)nis a natural number. .Prove that:

(a) 3^{2n} - 2^n is divisible by 7

(b) 5^{2n} - 2^n is divisible by 23

(a) We have: .N .= .(3²)^n - 2^n .= .9^n - 2^n .= .(7 + 2)^n - 2^n

Expand the binomial:

N .= .[7^n + n·7^{n-1}·2 + [n(n-1)/2]·7^{n-2}·2² + . . . + n·7·2^{n-1} + 2^n] - 2^n

And we have:

N .= .7^n + n·7^{n-1}·2 + [n(n-1)/2]·7^{n-2}·2² + ... + n·7·2^{n-1}

Sinceeverytermhas a factor of 7,Nis divisible by 7.

(b) We have: .N .= .(5²)^n - 2^n .= .25^n - 2^n .= .(23 + 2)^n - 2^n

Now proceed as in part (a).