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G4.3 Prime Factors, HCF and LCM

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01 — Explanation

How do I break a number down into its prime factors?

Use a factor tree: split the number into any two factors, circle any that are already prime, and keep splitting the non-prime ones until every branch ends in a circled prime. Multiply all the circled primes together (writing repeated factors using powers) to get the prime factor decomposition.

Example — decompose 525: $525 = 5 \times 105 \to 105 = 5 \times 21 \to 21 = 3 \times 7$ . Collecting the circled primes: $525 = 3 \times 5^2 \times 7$ .

Exam Tip

Always finish in index form ( $3^1 \times 5^2 \times 7^1$ ), not the expanded list ( $3\times5\times5\times7$ ) — the index form is what the next two skills (HCF and LCM) actually need.

02 — Explanation

How do I find the Highest Common Factor (HCF)?

Write both numbers in prime-factor index form, then multiply together only the primes common to both, each raised to the lowest power it appears with.

Example — HCF of 72 and 90: $72 = 2^3\times3^2$ , $90=2\times3^2\times5$ . Both share $2$ and $3$ . Lowest power of $2$ is $2^1$ ; lowest power of $3$ is $3^2$ . $\text{HCF}=2\times9=18$ .

The "lowest power" rule feels backwards at first, but it's logical: a common factor has to divide into both original numbers, so it can't contain more of a prime than the number with the least of it.

03 — Explanation

How do I find the Lowest Common Multiple (LCM)?

Take every distinct prime appearing in either number, each raised to the highest power it appears with anywhere, and multiply them together.

Example — LCM of 72 and 90: distinct primes are $2,3,5$ . Highest powers: $2^3$ , $3^2$ , $5^1$ . $\text{LCM}=8\times9\times5=360$ .

Exam Tip

Students often swap the two rules — "highest for HCF, lowest for LCM" — because of the words "Highest" and "Lowest" in the names. It's the opposite: the multiple must be big enough to contain everything (highest power), while the factor must be small enough to fit inside everything (lowest power).

A useful self-check: $\text{HCF} \times \text{LCM} =$ (first number) $\times$ (second number). For 72 and 90: $18 \times 360 = 6480 = 72 \times 90$ . ✓

04 — Explanation

How do I know whether a word problem needs HCF or LCM?

Clue in the question	What's needed
"Largest number of identical groups", "greatest number that divides into"	HCF
"When will they next coincide", "happen at the same time again"	LCM

Example: two buses leave every 24 and 40 minutes. When do they next leave together? This is a coincidence/timing question, so it's LCM. $24=2^3\times3$ , $40=2^3\times5$ . $\text{LCM}=2^3\times3\times5=120$ minutes — i.e. 2 hours later.

05 — Explanation

How do I secure method marks on these questions?

Showing at least two steps of a factor tree, or writing both numbers in index form before selecting the common/distinct primes, is enough to secure the method mark even if your final arithmetic slips. A final answer with no PFD working risks losing marks entirely if it's wrong.

06 — Worked example

Worked example

Find the LCM of $5A$ where $A = 2^3\times3\times5^2$ and $7B$ where $B=2^2\times3^3\times5$ .

$5A = 5\times2^3\times3\times5^2 = 2^3\times3\times5^3$ .
$7B = 7\times2^2\times3^3\times5 = 2^2\times3^3\times5\times7$ .
Take the highest power of each prime present anywhere: $2^3, 3^3, 5^3, 7^1$ .
$\text{LCM} = 8\times27\times125\times7 = 189\,000$ .

Treat the algebraic manipulation ( $5A$ , $7B$ ) as a first step, then apply the ordinary LCM rule exactly as before.