Factors & Factorisation

At a glance 11 formulas

1. 1Factorise firstWrite $N = p^a q^b r^c$ with $p, q, r$ prime. Every factor picks a power of each prime, from 0 up to its exponent.
2. 2Number of factors$N = p^a q^b r^c \Rightarrow (a+1)(b+1)(c+1)$
3. 3Two or three factorsExactly 2 factors: a prime. Exactly 3 factors: the square of a prime, such as $49 = 7^2$.
4. 4Odd number of factorsFactors pair up as $d$ and $\tfrac{N}{d}$, so the count is odd only for a perfect square.
5. 5Odd and even factors$N = 2^a q^b r^c:\ \text{odd} = (b+1)(c+1)$$\text{even} = \text{all} - \text{odd} = a\,(b+1)(c+1)$
6. 6Factors divisible by kThe factors of $N$ that are multiples of $k$ match the factors of $\tfrac{N}{k}$ one for one, so count those.
7. 7Square and cube factorsSquares allow only even powers: $p^a$ gives $\lfloor a/2 \rfloor + 1$ choices; for cubes, $\lfloor a/3 \rfloor + 1$. Multiply the choices across the primes.
8. 8Sum of factors$(1 + p + \dots + p^a)(1 + q + \dots + q^b)$$1 + p + \dots + p^a = \dfrac{p^{a+1} - 1}{p - 1}$
9. 9Factor pairs$f \text{ factors} \Rightarrow \dfrac{f}{2} \text{ pairs } x \times y = N$$\text{if } N \text{ is a square: } \dfrac{f + 1}{2} \text{ pairs}$
10. 10Smallest with f factorsSplit $f$ as $(a+1)(b+1)\dots$, put the biggest exponent on 2, the next on 3, and compare: 6 factors gives $2^2 \times 3 = 12$.
11. 11Product of factors$\text{product of all } f \text{ factors} = N^{f/2}$