The two ideas
HCF (highest common factor) is the largest number that divides all the given numbers. Think "biggest equal pieces".
LCM (lowest common multiple) is the smallest number that all the given numbers divide. Think "first time things line up again".
Prime factor method
Write each number as a product of prime powers.
HCF: the lowest power of each prime common to all numbers.
LCM: the highest power of every prime that appears.
Example. 36 = 2² × 3² and 84 = 2² × 3 × 7. HCF = 2² × 3 = 12. LCM = 2² × 3² × 7 = 252.
The division (Euclid) method is quicker for two large numbers: HCF(a, b) = HCF(b, a mod b), repeated until the remainder is 0.
The product rule
For any two numbers: HCF × LCM = product of the numbers.
Example. Two numbers have product 1080 and HCF 6. Their LCM is 1080 ÷ 6 = 180.
The rule does not hold for three or more numbers.
Which one is the question asking for?
Clue in the question | Use |
|---|---|
largest tile, longest rope, greatest number of equal groups | HCF |
bells ringing together, runners meeting, lights flashing together | LCM |
greatest number dividing several numbers leaving the same remainder | HCF of the differences |
least number leaving the same remainder r when divided by several numbers | LCM + r |
Example: same remainder. The greatest number dividing 43, 91 and 183 with the same remainder divides their differences 48, 92 and 140. HCF(48, 92, 140) = 4.
Example: bells. Bells toll every 9, 12 and 15 minutes. They toll together every LCM(9, 12, 15) = 180 minutes = 3 hours.
Numbers in a ratio
If two numbers are in the ratio a : b (in lowest terms) with HCF h, the numbers are ah and bh, and their LCM is abh.
Example. Numbers in ratio 3 : 4 with LCM 180: 12h = 180 gives h = 15, so the numbers are 45 and 60.
Fractions
HCF of fractions = HCF of numerators ÷ LCM of denominators
LCM of fractions = LCM of numerators ÷ HCF of denominators
Least perfect square divisible by several numbers
Find the LCM, then multiply by whatever is needed to make every prime power even.
Example. LCM(6, 8, 15) = 120 = 2³ × 3 × 5. Multiply by 2 × 3 × 5 = 30 to get 3600 = 60².
Shortcuts and tricks
Prime factorisation always works. These shortcuts save time when the numbers are large or the question has a familiar shape.
HCF by repeated subtraction (Euclid)
Example: the HCF of 391 and 323.
Standard way: factorise: 391 = 17 × 23 and 323 = 17 × 19, so the HCF is 17.
Shortcut: the HCF also divides the difference, so keep replacing the larger number with the remainder: HCF(391, 323) = HCF(323, 68) = HCF(68, 51) = HCF(51, 17) = 17.
LCM two at a time
Example: the LCM of 8, 12 and 20.
Standard way: 8 = 2³, 12 = 2² × 3 and 20 = 2² × 5. The highest powers give 2³ × 3 × 5 = 120.
Shortcut: LCM(8, 12) = 24 by inspection, then LCM(24, 20) = 24 × 20 ÷ HCF(24, 20) = 480 ÷ 4 = 120.
The product rule
Example: the HCF of two numbers is 12, their LCM is 360, and one number is 72. Find the other.
Standard way: 72 = 2³ × 3² and 360 = 2³ × 3² × 5, so the other number supplies the 5. The HCF 12 = 2² × 3 means it has exactly 2² and 3. The other number is 2² × 3 × 5 = 60.
Shortcut: HCF × LCM = product, so the other number is 12 × 360 ÷ 72 = 60.
Same remainder: use the differences
Example: the greatest number that divides 70, 110 and 190 leaving the same remainder.
Standard way: test large divisors one by one until all three remainders match.
Shortcut: the divisor must divide every difference: 40, 80 and 120. Their HCF is 40 (each number leaves 30).
Exam tips
The HCF always divides the LCM.
Convert units first in tile and rope problems (6 m 24 cm = 624 cm).
Check options: the HCF can't be larger than the smallest number.