Factors, multiples and primes
The words
- A factor divides exactly into a number. The factors of 12 are 1, 2, 3, 4, 6, 12. Find them in pairs: 1 × 12, 2 × 6, 3 × 4.
- A multiple is in a number's times table: 12, 24, 36, ...
- A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, ... 1 is not prime (it has one factor), and 2 is the only even prime.
- A square number is a whole number times itself: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... (7² = 7 × 7 = 49).
- A cube number is a whole number times itself three times: 1, 8, 27, 64, 125, ... (4³ = 4 × 4 × 4 = 64).
To find the primes in a range, cross out the even numbers (except 2), then the multiples of 3, 5 and 7. Between 30 and 50 that leaves 31, 37, 41, 43, 47.
Watch numbers like 49 = 7 × 7 and 51 = 3 × 17: odd, but not prime.
A product of prime factors
Keep dividing by primes, smallest first, until you reach 1.
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5
Writing 2 × 2 × 2 as 2³ is called index form: the small raised number (the index, or power) says how many times the prime is multiplied. Check by multiplying back: 8 × 9 × 5 = 360.
A factor tree gives the same primes; every number has exactly one prime factorisation.
HCF and LCM
The highest common factor is the biggest number that divides both.
The lowest common multiple is the smallest number both divide into.
For small numbers, list them.
- HCF of 18 and 24: factors of 18 are 1, 2, 3, 6, 9, 18 and factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24, so the HCF is 6.
- LCM of 6 and 10: multiples of 6 are 6, 12, 18, 24, 30 and multiples of 10 are 10, 20, 30, so the LCM is 30.
Multiplying the two numbers (6 × 10 = 60) gives a common multiple, but not always the lowest.
From prime factors: 60 = 2² × 3 × 5 and 72 = 2³ × 3².
- HCF: the primes in both, with the lower power: 2² × 3 = 12.
- LCM: every prime, with the higher power: 2³ × 3² × 5 = 360.
Problems about things that repeat (buses, lights, packs that must match) are LCM problems. Problems about splitting into the biggest equal groups are HCF problems.
- one bell rings every 12 minutes and another every 20 minutes; both ring at noon
- multiples of 12: 12, 24, 36, 48, 60
- multiples of 20: 20, 40, 60
- LCM = 60, so they next ring together 60 minutes after noon, at 1 pm
- cups come in packs of 6 and lids in packs of 8; you want the same number of each
- LCM of 6 and 8 = 24
- packs of cups: 24 ÷ 6 = 4
- packs of lids: 24 ÷ 8 = 3
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