KCSE 2025 Maths P1 Q6 — Highest Common Factor (Least Number of Families)

KCSE 2025 Form 4 Numbers

Published

The Question

“A relief organisation donated 240 kg of maize and 150 kg of beans to needy families. Each family received exactly the same quantity by mass of either maize or beans, and no family received both maize and beans. Determine the least possible number of needy families.”

1

Understand what gives the fewest families

Because every family receives the same mass and no family receives both foods, the mass each family gets must divide both 240 and 150 exactly. To make the number of families as small as possible, each family should receive the largest possible equal share. That largest share is the highest common factor of 240 and 150.

Bigger shares mean fewer families, so we want the greatest common divisor.

2

Write each number as a product of primes

Break 240 and 150 down into their prime factors so we can see exactly which factors they have in common. This lets us build the HCF instead of guessing.

240=24×3×5240 = 2^{4} \times 3 \times 5
150=2×3×52150 = 2 \times 3 \times 5^{2}
3

Build the HCF from the shared primes

The highest common factor uses only the primes that appear in both numbers, and takes each one at its lowest power. Both numbers contain one 2, one 3 and one 5, so we multiply those together.

HCF=2×3×5=30\text{HCF} = 2 \times 3 \times 5 = 30

So the largest equal share possible is 30 kg per family.

4

Divide each donation by the share

Now split each type of food into 30 kg portions. The number of maize families plus the number of beans families gives the total number of families.

24030=8(maize families)\frac{240}{30} = 8 \quad \text{(maize families)}
15030=5(beans families)\frac{150}{30} = 5 \quad \text{(beans families)}
8+5=138 + 5 = 13

Final Result

The least possible number of needy families is 13 — eight families sharing the maize and five families sharing the beans, each receiving 30 kg.

Why this method works

Using the HCF works because the share must divide both 240 and 150 without remainder, so it has to be a common factor of the two masses. The number of families is inversely related to the size of each share, so the fewest families come from the biggest allowed share, which is the greatest of those common factors — the highest common factor. Any smaller common factor, such as 15 or 10, would break the maize and beans into more portions and therefore need more families.

✓

Check: 30 kg divides both amounts exactly since 8 times 30 is 240 and 5 times 30 is 150, confirming 13 families in total.