Prime factorisation, HCF and LCM

Writing numbers as products of prime factors, and using prime factorisation to find the highest common factor and lowest common multiple of large numbers.

Prime factorisation

Divide by the smallest prime that works, again and again, until you reach $1$. For example, $84 = 2 \times 42 = 2 \times 2 \times 21 = 2^2 \times 3 \times 7$.

HCF and LCM from primes

Example 1●●

Find the HCF and LCM of $84$ and $90$.

$84 = 2^2 \times 3 \times 7$ and $90 = 2 \times 3^2 \times 5$.

HCF: shared primes $2$ and $3$, lowest powers: $2 \times 3 = 6$.

LCM: $2^2 \times 3^2 \times 5 \times 7 = 1260$.

Example 2●●

Tiles are $24$ cm by $36$ cm. What is the smallest square that can be made from whole tiles, all placed the same way?

The side must be a common multiple of $24$ and $36$. LCM $= 72$.

A $72$ cm square, using $3 \times 2 = 6$ tiles.

Flashcards

HCF from primes

Shared primes, lowest powers.

LCM from primes

All primes, highest powers.

$84$ as primes

$2^2 \times 3 \times 7$

Practice

Find the HCF of $84$ and $90$.
Find the LCM of $84$ and $90$.
Find the LCM of $24$ and $36$.
Find the HCF of $24$ and $36$.
$2^3 \times 3 \times 5$ equals:
$126$ as a product of primes is:
Find the HCF of $2^2 \times 3 \times 5$ and $3^2 \times 5 \times 7$.
Two buses leave together. One returns every $12$ minutes, the other every $18$. After how many minutes are they together again?