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Answer:

Let P be any prime number. As P is a prime number,the only positive divisors of P are 1,P.

Similarly ,the only positive divisors of P ^2 are 1,P,P^2.Likewise the only positive divisors of 

P^3 are 1,P, P^2, P^3 and so on for other higher powers of P. That is for any positive integer

Power of P,the  only positive divisors   are 1,P and all higher powers of P

Let m be any positive integer.  Then the only positive divisors of P^m = 1,P,P^2,P^3……..&P^m. 

Therefore  σ (P^m) = sum of all the positive divisors of P^m = 1+P+P^2 +P^3+.......+P^m=

1+P( 1+P+P^2+..............P^m-1).

But 1+P+P^2+...........P^m-1 =  sum of all the positive divisors of P^m-1 = σ(P^m-1)

Hence we get σ(P^m) = 1+P*σ(P^m-1)

 

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