When one number divides the other

The idea

The HCF of two numbers is one of the numbers when that number is a factor of the other (the other being its multiple); in letters, the HCF of n and any multiple of n is n.

6 and 18

The HCF of 6 and 18 is 6: one of the two numbers! The book asks: find more pairs like this.

Why do you think the HCF of 6 and 18 is 6 itself?

• 6 divides 18
• 18 divides 6
• Both are even

Hunt for pairs

Pick two numbers. The toy shows their HCF. Record pairs in the right column, and see what the pairs of the first column have in common.

The smaller divides the bigger

In every pair of the first column, the bigger number is the smaller one times a whole number: 18 = 3 × 6. The smaller is a factor of the bigger, and the bigger is its multiple.

In letters: any multiple of a number n is a whole number times n, such as 5n. The HCF of n and 5n is n.

Why? Every factor of n is a factor of 5n too, and no factor of n is bigger than n. So n is the highest common factor.

The HCF of two numbers is one of the numbers when that number is a factor of the other (the other is its multiple). In letters, the HCF of n and any multiple of n is n.

• n | 5n | HCF
• 4 | 20 | 4
• 7 | 35 | 7
• 12 | 60 | 12

Notes

The HCF of two numbers is one of the numbers when that number is a factor of the other. In letters, the HCF of n and any multiple of n is n.

Check yourself

Find the HCF of 9 and 63.

Answer: 9

63 = 7 × 9, so 9 is a factor of 63. The HCF of 9 and 63 is 9.

In which pair is the HCF one of the two numbers?

A pair of numbers is n and 6n. Take n = 11. What is the HCF of the two numbers?

Answer: 11

The numbers are 11 and 66. 66 = 6 × 11, so the HCF is 11, which is n.

The HCF of two numbers is the smaller number. What can you say about the bigger number?

• 8 and 12. 12 is not a multiple of 8. The HCF of 8 and 12 is 4, neither of them.
• 9 and 15. 15 is not a multiple of 9. The HCF of 9 and 15 is 3, neither of them.
• 8 and 24 — correct. Yes! 24 = 3 × 8, so 8 is a factor of 24 and the HCF is 8.
• It is a multiple of the smaller number. — correct. Yes! If the smaller is a common factor, it divides the bigger one, so the bigger is a multiple of it.
• It is a factor of the smaller number.. A factor cannot be bigger than the number. The bigger number cannot divide the smaller.
• It is exactly double the smaller number.. It could be double, but it could also be 3, 5 or 7 times the smaller (6 and 18, 7 and 35).