Let n exist in Z with n0 Prove that that the sum of positive

Let n exist in Z with n>0. Prove that that the sum of positive divisors function theta is n<= theta(n) <=n^2.

Solution

Let n be an integer >0

If n has positive divisors 1 and n always

If n is composite it may have other divisors also

sum of all divisors =n+1 hence >n

and n(n) is always less than n(products involving less than n)

Hence n<= sum of all divisors<= n(n)