6. Number of Factors and Sum of Factors

Figure: Break a number down to its prime factors with a factor tree.

For a composite number N, first express it in prime factorized form: N = a^p × b^q × c^r × … (where a, b, c are distinct primes)

Number of factors (including 1 and N itself): Number of factors = (p+1)(q+1)(r+1)…

Sum of all factors: Sum of factors = [(a^(p+1) − 1)/(a − 1)] × [(b^(q+1) − 1)/(b − 1)] × [(c^(r+1) − 1)/(c − 1)] × …

Why it works: Any factor of N must be of the form a^i × b^j × c^k… where 0 ≤ i ≤ p, 0 ≤ j ≤ q, etc. There are (p+1) choices for i, (q+1) for j, and so on — multiplying these choices (fundamental counting principle) gives the total count. The sum formula is simply the expansion of (1+a+a²+…+aᵖ)(1+b+…+b^q)… using the geometric series sum formula, which multiplies out to give the sum of every possible combination.