Problem 7. For each positive integer n, define 2(n) to be the sum of the squares of all positive divisors of n. As an example, 2(6) = 1^2 + 2^2 + 3^2 + 6^2 = 50 Prove that 2 : N N defined as above is a multiplicative function, but it is not a completely multiplicative function. Remark. By definition, a multiplicative function f must satisfy f(mn) = f(m)f(n) whenever gcd(m, n) = 1. In contrast, a completely multiplicative function f must satisfy f(mn) = f(m)f(n) for all positive integers m and n. It is clear that every completely multiplicative function is multiplicative.

Problem 8. Referring to Problem 7, compute 2(n) for all n N. Your final answer will depend on the prime factorization of the input number n. Use the formula you developed to compute 2(168).