Factors, Multiples, HCF and LCM

In short: A factor divides a number exactly, and a multiple is in its times table. The highest common factor (HCF) is the biggest factor two numbers share, and the lowest common multiple (LCM) is the smallest multiple they share. Use the LCM when repeating things meet again.

Before you start

• Times tables up to $12\times 12$.
• Division with remainders (see Long Multiplication and Long Division).

Factors and multiples

Factors come in pairs that multiply to the number: $1\times 12$, $2\times 6$, $3\times 4$.

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... The number 1 is not prime, and 2 is the only even prime.

Divisibility tests

Odd and even

If 41 drinks are apple or mango, the two amounts cannot differ by 8. Two numbers that add to an odd number are one odd and one even, so their difference is odd too.

Remainders

When 44 is divided by 5, the remainder is 4, because $44=8\times 5+4$. Numbers with the same remainder when divided by 5 end in the same digit, or in digits 5 apart: 29, 44, 59 and 74 all leave 4.

An odd number divided by 6 can only leave an odd remainder: 1, 3 or 5.

Highest common factor (HCF)

List the factors of each number and pick the biggest one in both lists.

Lowest common multiple (LCM)

List multiples of the bigger number until one is also a multiple of the smaller.

Quick summary

• A factor divides a number exactly. A multiple is in its times table.
• A prime has exactly two factors. 1 is not prime, and 2 is the only even prime.
• Divisibility by 3 or 9: add the digits. By 4: check the last two digits.
• Odd $+$ odd is even, and only odd $\times$ odd is odd.
• HCF is the biggest shared factor, LCM the smallest shared multiple.
• Things that repeat meet again after the LCM.

Quiz