Patterns and general statements

The HCF of 6 and 18 is 6, which is one of the two given numbers. Find more pairs where the HCF is one of the numbers — 4 and 12, 7 and 35, 9 and 9. How can we describe such pairs?

It happens exactly when one number is a FACTOR of the other. And that is the same as saying the other number is a MULTIPLE of the first. 'Be my factor, I'll be your multiple.'

A statement describing a pattern or property that holds in ALL possible cases is called a general statement, and the process of finding one is GENERALISATION. It is a step up from noticing: it claims something about every case at once.

Algebra says it compactly. If n is a number, any multiple of n can be written as a positive integer times n. So take n and 5n: then 5n is a multiple of n, and n is a factor of 5n. The HCF of n and 5n is n. If one of the numbers is m, the other could be any multiple of m; if one is 7k, the other could be any multiple of 7k.

Try making general statements of your own for these pairs, testing examples first: two consecutive even numbers, two consecutive odd numbers, two even numbers, two consecutive numbers, two co-prime numbers. And the same for the LCM.

Here is another pattern. What happens to the HCF of two numbers if BOTH are doubled? Try 270 and 50: 270 = 2 × 3 × 3 × 3 × 5 and 50 = 2 × 5 × 5, so the HCF is 2 × 5 = 10. Double them to 540 and 100: 540 = 2 × 2 × 3 × 3 × 3 × 5 and 100 = 2 × 2 × 5 × 5, so the HCF is 2 × 2 × 5 = 20. The HCF doubled too.

And the reason is visible in the factorisations. Doubling gives each number an extra factor of 2, so that extra 2 is now common, and it joins the largest common subpart. The same reasoning handles two multiples of one number: for 14 × 6 = 2 × 7 × 2 × 3 and 14 × 9 = 2 × 7 × 3 × 3, the HCF is 14 × 3 = 42 — the common multiplier 14 and whatever else 6 and 9 share.