Key ideas
- Multiples of a number are its times table: the multiples of 4 are 4, 8, 12, 16, …
- Factors of a number divide into it exactly: the factors of 12 are 1, 2, 3, 4, 6 and 12. Find them in pairs.
- The lowest common multiple (LCM) is the smallest number in both times tables. The highest common factor (HCF) is the biggest number that divides into both.
- A prime number has exactly two factors: 1 and itself. 1 is not prime. 2 is the only even prime.
- Prime decomposition writes a number as primes multiplied together. Use a factor tree.
- Powers: 5² = 5 × 5 = 25 and 2³ = 2 × 2 × 2 = 8. Roots undo them: 25 = 5 and 8 = 2.
Multiples
List the first six multiples of 7.(1 mark)
Go through the 7 times table.7 × 1, 7 × 2, 7 × 3, 7 × 4, 7 × 5, 7 × 6
In plain EnglishMultiples are the answers in a times table. Start with 7 × 1 (the number itself counts as a multiple).
Write them out.7, 14, 21, 28, 35, 42✓1
In plain EnglishEach time, add another 7.
⚠ Don't confuse them. Multiples get bigger (7, 14, 21, …). Factors are smaller and divide in.
Lowest common multiple
Hot dogs come in packs of 6 and rolls in packs of 8. What is the smallest number of hot dogs you can buy to have exactly the same number of rolls? (Find the LCM of 6 and 8.)(2 marks)
List multiples of each number.6: 6, 12, 18, 24, 30, 368: 8, 16, 24, 32, 40✓1
In plain EnglishWrite out each times table. Stop once you spot a number that is in both lists.
Pick the first number in both lists.LCM = 24✓2
In plain English24 is the first number that appears in both lists. 48 is also in both, but 24 is the lowest.
Factors
List all the factors of 24.(2 marks)
Find the factor pairs.
| 1 | × | 24 |
| 2 | × | 12 |
| 3 | × | 8 |
| 4 | × | 6 |
In plain EnglishStart with 1 × 24. Then try 2, 3, 4, … Each time a number divides in exactly, write down the pair. 5 doesn't go into 24, so skip it.
Stop when the pairs meet.5 × ? ✗ (5 doesn't go into 24)6 is already in a pair → stop✓1
In plain EnglishOnce the numbers you try reach a number you've already written (6), you have every pair.
Write them in order.1, 2, 3, 4, 6, 8, 12, 24✓2
In plain EnglishRead down the left column, then up the right column.
Tip. Factors always come in pairs. Working in pairs means you won't miss any.
Highest common factor
Find the highest common factor of 18 and 30.(2 marks)
List the factors of each number.18: 1, 2, 3, 6, 9, 1830: 1, 2, 3, 5, 6, 10, 15, 30✓1
In plain EnglishUse factor pairs for each number so you don't miss any.
Find the biggest one in both lists.Common factors: 1, 2, 3, 6HCF = 6✓2
In plain English1, 2, 3 and 6 are in both lists. The highest of these is 6.
Prime numbers
Which of these numbers are prime? 21, 23, 25, 27, 29(2 marks)
Check each number for other factors.21 = 3 × 7 ✗23: only 1 × 23 ✓25 = 5 × 5 ✗27 = 3 × 9 ✗29: only 1 × 29 ✓✓1
In plain EnglishA prime number has exactly two factors: 1 and itself. Try dividing by small primes (2, 3, 5, 7). If one goes in, it isn't prime.
Write the primes.23 and 29✓2
In plain English23 and 29 have no factors except 1 and themselves.
⚠ Remember. 1 is not prime (it only has one factor). 2 is prime, and it's the only even prime.
Prime decomposition
Write 60 as a product of its prime factors.(3 marks)
Split 60 into any factor pair.60 = 6 × 10✓1
In plain EnglishStart the factor tree with any pair that multiplies to 60. 6 × 10 is a good choice.
Keep splitting until every branch ends in a prime.6 = 2 × 3 10 = 2 × 5✓2
In plain English6 and 10 aren't prime, so split them again. 2, 3, 2 and 5 are all prime, so circle them. The tree stops there.
Multiply the circled primes.60 = 2 × 2 × 3 × 5✓3= 22 × 3 × 5
In plain EnglishWrite the circled primes from smallest to largest with × between them. You can write 2 × 2 as 2² (index form).
Check: 2 × 2 × 3 × 5 = 60 ✓. Whichever pair you start with, you always end with the same primes.
Powers and roots
Work out (a) 5² (b) 2³ (c) 49 (d) 27(4 marks)
(a) Squared means × itself.5² = 5 × 5 = 25✓1
In plain EnglishThe small 2 is the power. It tells you how many 5s to multiply. It does not mean 5 × 2.
(b) Cubed means three of them.2³ = 2 × 2 × 2 = 8✓2
In plain English2 × 2 = 4, then 4 × 2 = 8.
(c) Square root: what times itself?49 = 7 because 7 × 7 = 49✓3
In plain EnglishA square root undoes squaring. Which number multiplied by itself gives 49?
(d) Cube root: what × itself × itself?27 = 3 because 3 × 3 × 3 = 27✓4
In plain EnglishWhich number, used three times, multiplies to 27?
Worth learning: square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and cube numbers 1, 8, 27, 64, 125.
0 of 0 complete
Your turn
Non-calculator. For lists, separate the numbers with commas. Stuck? Tap Need a hint?