Tutor lessons
HCF & LCM ladder
One neat trick finds both the Highest Common Factor and the Lowest Common Multiple: keep dividing both numbers by a shared factor until nothing more is shared. The left column builds the HCF; the LCM drops out at the end.
Goal
Build the ladder one row at a time. Stop only when the last two numbers are coprime — then read off the HCF and LCM.
Learn the ladder method
Divide both numbers by common factors until they are coprime. The left column gives the HCF. For the LCM, divide the first number by the HCF, then multiply by the second number.
HCF
Highest Common Factor — the biggest number that divides exactly into both numbers.
LCM
Lowest Common Multiple — the smallest number that both numbers divide into exactly.
Common factor
A number greater than 1 that divides both numbers with nothing left over. The ladder keeps peeling these off.
Worked example: HCF and LCM of 60 and 84
Divide both numbers by a common factor over and over. Each divisor goes in the left column. Stop when the two numbers left share no factor bigger than 1 — here that is 5 and 7.
HCF = 2 × 2 × 3 = 12
Multiply the divisors down the left column.
LCM shortcut
60 ÷ 12 = 5
5 × 84 = 420
Divide the first number by the HCF, then multiply by the second number.
Why does this method work?▾
Every divisor you take out is a factor shared by both numbers, so the product of the left column is exactly the highest common factor.
Once the two numbers left are coprime, nothing more is shared. The LCM has to contain the shared part once (the HCF) plus whatever is left over in each number.
The ladder shows the longer version: 12 × 5 × 7 = 420. In the shortcut, 60 ÷ 12 recovers the leftover 5. The second number, 84, already contains the shared 12 and the other leftover 7, so multiplying them gives the same answer.
Important: keep going until the numbers are coprime▾
You are only finished when the two remaining numbers have an HCF of 1 (they are coprime, like 3 and 5). If they still share a factor, you have stopped too early and your HCF will be too small.
This is the “divides BOTH” ladder▾
A divisor is only allowed if it divides both numbers exactly. This is different from the other ladder method where a divisor may divide only one of the numbers — do not mix them up.
Practise
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