60-second revision
- HCF = product of common primes with the lowest powers; LCM = all primes with the highest powers.
- For two numbers only: HCF × LCM = product of the numbers.
- HCF of fractions = HCF of numerators / LCM of denominators; LCM of fractions = LCM of numerators / HCF of denominators.
- Least number leaving the same remainder r with divisors a, b, c = LCM(a, b, c) + r.
- Greatest number dividing x, y, z leaving the same remainder r = HCF(x - r, y - r, z - r).
- Greatest number dividing x, y, z leaving the same (unknown) remainder = HCF of the differences.
- "Together again" (bells, lights, runners) is an LCM question; "largest equal piece or tile" is an HCF question.
Definitions
HCF (highest common factor, also GCD) is the largest number that divides each given number exactly. LCM (least common multiple) is the smallest number that each given number divides exactly. The HCF always divides the LCM.
Prime factorisation method
Worked example: 96 = 2^5 × 3, 144 = 2^4 × 3², 240 = 2^4 × 3 × 5. HCF takes only primes common to all, with the lowest powers: 2^4 × 3 = 48. LCM takes every prime with the highest power: 2^5 × 3² × 5 = 1,440.
Division (Euclid) method for HCF
Divide the larger number by the smaller, then divide the divisor by the remainder, and repeat until the remainder is 0. The last divisor is the HCF. Example: HCF of 1,080 and 1,440. 1,440 = 1 × 1,080 + 360; 1,080 = 3 × 360 + 0. HCF = 360.
Product rule (two numbers only)
HCF × LCM = first number × second number. For 1,080 and 1,440: HCF 360 and LCM 4,320, and 360 × 4,320 = 1,080 × 1,440. If two numbers have HCF h, write them as ha and hb with a, b co-prime; then LCM = hab. Example: numbers in ratio 2:5 with LCM 280: 10h = 280, h = 28, numbers 56 and 140.
LCM and HCF of fractions
| Find | Formula | Example |
|---|---|---|
| HCF of fractions | HCF of numerators / LCM of denominators | HCF of 4/5, 6/7, 10/9 = 2/315 |
| LCM of fractions | LCM of numerators / HCF of denominators | LCM of 2/3, 4/9, 5/6 = 20/3 |
Write each fraction in lowest terms before using the formula.
Remainder problems: the four standard types
| Question asks for | Method | Worked example |
|---|---|---|
| Least number leaving the same remainder r | LCM + r | Divided by 9, 12, 15 leaves 4 each time: 180 + 4 = 184 |
| Least number leaving remainders that each fall short of the divisor by d | LCM - d | Divided by 4, 5, 6 leaves 1, 2, 3 (each short by 3): 60 - 3 = 57 |
| Greatest number dividing x, y leaving remainder r | HCF(x - r, y - r) | 150 and 222 leaving 6: HCF(144, 216) = 72 |
| Greatest number dividing x, y, z leaving the same unknown remainder | HCF of the differences | 31, 67, 103, 139: differences 36, 36, 36, HCF 36 |
Variant: "greatest number dividing 70 and 95 leaving remainders 5 and 4 respectively" means HCF(70 - 5, 95 - 4) = HCF(65, 91) = 13.
Word problems
- Bells, lights, runners on a track meeting at the start: LCM of the intervals. Convert to a common unit (seconds) first.
- Cutting rods or ribbons into the longest equal pieces: HCF of the lengths.
- Least number of square tiles for a floor: tile side = HCF of the sides. A 504 cm × 360 cm floor uses 72 cm tiles, and (504/72) × (360/72) = 7 × 5 = 35 tiles.
- Smallest square or perfect square divisible by given numbers: take the LCM, then raise every odd prime power to the next even power.
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