Prime factorisation

Prime factorisation means writing a whole number as a product of prime numbers — numbers whose only factors are $1$ and themselves.
Every whole number greater than $1$ has exactly one such prime breakdown.

The easiest method is a factor tree.
Split the number into any two factors, then keep splitting each factor until you are left only with primes.

Once you reach the primes, group the repeats using exponents.
For example, breaking down $80$ gives $2, 2, 2, 2$, and $5$, which you write as $80 = 2^4 \times 5$.
Writing it this way is compact and easy to work with.

Prime factorisation is the key to many number questions.
Once you have the primes, you can list every factor, check divisibility, or find a greatest common factor.
For instance, $930 = 2 \times 3 \times 5 \times 31$, so any product of those primes is a factor of $930$.

What is the prime factorisation of $72$?
Break it down: $72 = 8 \times 9 = (2 \times 2 \times 2) \times (3 \times 3)$.
Grouping the repeats, $72 = 2^3 \times 3^2$.

What is the prime factorisation of $5400$?
Break it down: $5400 = 54 \times 100 = (2 \times 27) \times (4 \times 25) = 2 \times 3^3 \times 2^2 \times 5^2$.
Grouping the repeats, $5400 = 2^3 \times 3^3 \times 5^2$.

Given that $84 = 2^2 \times 3 \times 7$, is $21$ a factor of $84$?
Since $21 = 3 \times 7$, and both $3$ and $7$ appear in the prime factorisation of $84$, their product $21$ divides it.
So yes, $21$ is a factor of $84$.