# Factors, Multiples, HCF and LCM

> Find factors and multiples, use divisibility tests, work with odd and even numbers and remainders, and find the HCF and LCM of two or three numbers.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/factors-and-multiples/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The flashing lights
Three lights on a stage flash at different speeds. The red one flashes every 4 seconds and the green one every 6 seconds. They both flash at the start of the show.

"When do they flash together again?" asks Tom. He writes the red times: 4, 8, 12, 16, 20, 24. Then the green: 6, 12, 18, 24. "12 seconds!" he says, "and again at 24."

That first shared time, 12, is called the lowest common multiple. Questions about things that repeat and meet again are really questions about multiples.

**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

### Key idea: Two words
- A **factor** of 12 divides into 12 exactly: 1, 2, 3, 4, 6 and 12.
- A **multiple** of 12 is in the 12 times table: 12, 24, 36, 48, ...

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

| A number divides by | if |
|---|---|
| 2 | its last digit is even |
| 3 | its digits add up to a multiple of 3 |
| 4 | its last two digits make a multiple of 4 |
| 5 | it ends in 0 or 5 |
| 6 | it divides by both 2 and 3 |
| 9 | its digits add up to a multiple of 9 |
| 10 | it ends in 0 |

### Worked example: Using the tests
Does 2148 divide by 3? $2 + 1 + 4 + 8 = 15$, and 15 is a multiple of 3, so yes.

The largest 4-digit number that divides by both 4 and 5 divides by 20, so it ends in 00, 20, 40, 60 or 80: it is 9980.

## Odd and even

### Key idea: Odd and even rules
- even $+$ even $=$ even, odd $+$ odd $=$ even, odd $+$ even $=$ odd.
- even $\times$ anything $=$ even. Only odd $\times$ odd $=$ odd.

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.

### Worked example: HCF of 18 and 24
Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. The HCF is 6.

## Lowest common multiple (LCM)

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

### Worked example: LCM of three numbers
Three swimmers finish a lap every 6, 9 and 15 minutes. Multiples of 15: 15, 30, 45, 60, 75, 90. The first that 6 and 9 both divide is 90. They are next at the wall together after 90 minutes.

### Exam tip: HCF times LCM
For two numbers, HCF $\times$ LCM $=$ the two numbers multiplied together. For 6 and 8: HCF 2, LCM 24, and $2 \times 24 = 48 = 6 \times 8$.

### Challenge: Selective level: puzzles with factors
Two numbers have HCF 6 and LCM 60, and neither is 6 or 60. Both are multiples of 6 that divide into 60: 6, 12, 30 or 60. The pair must multiply to $6 \times 60 = 360$, so it is 12 and 30.

Bells ring every 2, 5 and 7 minutes, all together at 9:00. The 5 and 7 minute bells meet every 35 minutes, but every second meeting (70, 140, ...) is even, so the 2 minute bell rings too. The 5 and 7 minute bells ring without it at 9:35, 10:45, 11:55: every 70 minutes.

**Mind map**

```text
Factors and multiples
  Factors
    Divide exactly
    Come in pairs
    Primes have two
  Multiples
    The times table
  Tests
    3 and 9: add the digits
    4: last two digits
    5 and 10: last digit
  Odd and even
    Odd + odd = even
    Odd x odd = odd
  HCF and LCM
    HCF: biggest shared factor
    LCM: first shared multiple
    Repeating things meet at the LCM
```

## 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

Try these questions from the chapter quiz.

1. Which number is a factor of 24?
   - A. 5
   - B. 7
   - C. 8
   - D. 48
   - Answer: C. 8. $24 \div 8 = 3$ exactly. 48 is a multiple of 24, not a factor.

2. Which of these is a prime number?
   - A. 1
   - B. 9
   - C. 13
   - D. 21
   - Answer: C. 13. 13 has only two factors, 1 and 13. $9 = 3 \times 3$ and $21 = 3 \times 7$.

3. What is the remainder when 44 is divided by 5?
   - A. 2
   - B. 3
   - C. 4
   - D. 8
   - Answer: C. 4. $8 \times 5 = 40$, and $44 - 40 = 4$.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/factors-and-multiples/#quiz