Reading a Prime Factorisation

What a Prime Is

A prime number has exactly two factors, itself and 1. Two is the smallest, and one is not prime. Every whole number greater than 1 can be written as a unique product of primes, which is why factorisation is possible at all.

Notation

360 factorises as 2 cubed times 3 squared times 5, because 8 times 9 times 5 is 360. The superscript is a power, so 2 cubed means three 2s multiplied together.

Building One

Divide by 2 as often as possible, then by 3, then 5, and continue upward through the primes. Record how many times each divides, and those counts are the powers.

The GCD From Factorisations

For each prime take the smaller of the two powers. For 12 and 18, that is 2 to the power of 1 and 3 to the power of 1, giving a GCD of 6.

The LCM From Factorisations

For each prime take the larger of the two powers. For 12 and 18, that is 2 squared and 3 squared, which is 36. This is usually the fastest route for numbers with obvious structure.

Counting Divisors

The number of divisors of a number is found by multiplying each power plus one. For 12, which is 2 squared times 3, that is 3 times 2, giving 6 divisors.