How many numbers less than 100 have the sum of factors as odd?
Answer is 16
This question and explanation is taken from careerbless.com The link given derives the answer using some properties of number. But, I am trying to solve it in a different way, using a general formula to find sum of the factors of numbers, possibility that may lead to a simple solution.
I have gone through the post in this website where a formula is explained to find sum of the factors of a number. But, it cannot be applied individually for this question (because we need to do it for 1-99 and it takes lot of time).
Is there any shortcut method to derive the answer for this question, possibility using a formula to understand if the sum of the factors is even or odd? Please comment with different approaches for this problem.
Best Answer
For an odd prime, the sum of divisors of $p^k$ is odd if and only if $k$ is even, so that $p^k$ is an (odd) square.
However, $2^k$ has an odd sum of divisors for any $k \geq 0.$
Your numbers are: $$ 1,9,25,49,81, $$ $$ 2,18,50,98, $$ $$ 4,36, $$ $$ 8,72, $$ $$ 16, $$ $$ 32, $$ $$ 64. $$
I get 16.