KJSEA 2025 Maths Q2 — Highest Common Factor (HCF) of 120, 180 and 24

KJSEA 2025 Grade 9 Numbers

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The Question

“A school has 120 boys, 180 girls and 24 teachers. They are to be arranged into groups so that every group has the same number of boys, the same number of girls and the same number of teachers, with none left over. What is the largest number of such groups that can be formed?”

1

Recognise that this is an HCF problem

Each group must contain a whole number of boys, girls and teachers with nothing left over. That means the number of groups has to divide 120, 180 and 24 exactly. To make the groups as many as possible we need the biggest number that divides all three, which is their highest common factor.

More groups means fewer people per group, so the largest number of groups comes from the HCF, not the LCM.

2

Write each number as a product of prime factors

Break every number down into primes so we can see exactly which factors they share. Keep going until only prime numbers remain.

120=23×3×5120 = 2^{3} \times 3 \times 5
180=22×32×5180 = 2^{2} \times 3^{2} \times 5
24=23×324 = 2^{3} \times 3
3

Take the lowest power of each shared prime

For the HCF we only keep primes that appear in all three numbers, and for each of those we take the smallest power that appears. The prime 2 is shared, and its lowest power is two squared. The prime 3 is shared, and the lowest power is a single three. The prime 5 is missing from 24, so it is dropped.

HCF=22×3\text{HCF} = 2^{2} \times 3
4

Multiply to get the HCF

Multiply the chosen prime factors together to find the highest common factor, which is the largest number of equal groups.

22×3=4×3=122^{2} \times 3 = 4 \times 3 = 12

Final Result

The largest number of groups that can be formed is 12 (option A).

Why this method works

The number of groups must divide each of the three totals exactly, so it has to be a common factor of 120, 180 and 24. Any common factor gives a valid grouping, but a smaller factor makes fewer, larger groups while a larger factor makes more, smaller groups. The single largest value that still divides all three at once is the highest common factor, so choosing the HCF guarantees the maximum possible number of equal groups. Prime factorisation makes this exact, because a factor common to every number can only be built from primes that each number contains, and only up to the smallest power available.

✓

12 divides each total exactly: 120 ÷ 12 = 10 boys, 180 ÷ 12 = 15 girls and 24 ÷ 12 = 2 teachers per group, with none left over.