What HCF and LCM are
The highest common factor (HCF), also called the greatest common divisor, of two or more numbers is the largest number that divides all of them exactly.
The lowest common multiple (LCM) is the smallest number that all of them divide into exactly.
An easy sense check: the HCF is never larger than the smallest number given, and the LCM is never smaller than the largest number given. If one number divides another, the HCF is the smaller and the LCM is the larger.
Finding HCF and LCM
By prime factorisation. Write each number as a product of prime powers.
- HCF: for each prime that appears in every number, take the lowest power it has anywhere, and multiply those together.
- LCM: for every prime that appears in any number, take the highest power it has anywhere, and multiply those together.
Example: 12 equals 2 squared times 3; 18 equals 2 times 3 squared; 24 equals 2 cubed times 3. Primes common to all are 2 and 3, each to the lowest power 1, so the HCF is 2 times 3, which is 6. For the LCM take 2 cubed and 3 squared, giving 8 times 9, which is 72.
By division. HCF by successive division (the Euclidean method): divide the larger number by the smaller; then divide the divisor by the remainder; repeat until the remainder is 0 — the last non-zero divisor is the HCF. For more than two numbers, chain it. LCM by common division: write the numbers in a row, divide repeatedly by a prime that divides at least one of them (carrying the rest down), and stop when the last row is all 1s; the product of every divisor used is the LCM.
The product relation, and fractions
For exactly two numbers: HCF times LCM equals the product of the two numbers. So if you know three of the four, you can find the fourth. This does not extend to three or more numbers.
For fractions:
- HCF of fractions equals HCF of the numerators divided by LCM of the denominators.
- LCM of fractions equals LCM of the numerators divided by HCF of the denominators.
Reduce each fraction to lowest terms first.
HCF or LCM — which does the problem want?
- HCF when you are cutting or grouping into the largest equal parts: the largest tile that fits a floor exactly, the greatest length that measures several rods, the largest number of identical gift bags from different quantities of items.
- LCM when you are asking when events coincide again or the smallest quantity that fits several groupings: bells that toll together, lights that blink together, the least number of sweets that can be shared equally among 6, 8 or 12 children.
- "Largest that divides" points to HCF; "smallest that is divisible" points to LCM.
Worked example: HCF and LCM of a set
Find the HCF and LCM of 12, 18 and 24.
Prime factorise: 12 equals 2 squared times 3; 18 equals 2 times 3 squared; 24 equals 2 cubed times 3.
HCF: primes in all three are 2 and 3, each to the lowest power 1. HCF equals 2 times 3, which is 6.
LCM: highest powers anywhere are 2 cubed and 3 squared. LCM equals 8 times 9, which is 72.
Check: 72 divided by 12 is 6, by 18 is 4, by 24 is 3 — all whole numbers.
Worked example: tolling bells
Three bells toll at intervals of 6, 8 and 12 seconds. If they toll together at the start, how many more times do they toll together in the next hour?
They toll together every LCM of 6, 8 and 12 seconds. 6 equals 2 times 3; 8 equals 2 cubed; 12 equals 2 squared times 3. LCM equals 2 cubed times 3, which is 24 seconds.
One hour is 3,600 seconds. The number of 24-second gaps in 3,600 seconds is 3,600 divided by 24, which is 150.
So they toll together 150 more times in the hour (151 including the start).
Worked example: the product relation
Two numbers have HCF 13 and LCM 546. One of the numbers is 78. Find the other.
For two numbers, HCF times LCM equals their product. So the product is 13 times 546, which is 7,098.
The other number is 7,098 divided by 78, which is 91.
Check: HCF of 78 and 91 is 13 (78 equals 13 times 6, 91 equals 13 times 7), and LCM is 13 times 6 times 7, which is 546.
Common mistakes
- Using HCF times LCM equals product for three or more numbers. It only holds for two.
- Taking the highest power for HCF or the lowest for LCM. HCF uses the lowest power of the common primes; LCM uses the highest power of all primes.
- Forgetting that HCF needs a prime to appear in every number. A prime in only one number contributes to the LCM but not the HCF.
- Fraction rule inverted. HCF of fractions is HCF(numerators) over LCM(denominators); the LCM rule swaps them.
- Picking HCF when the question asks when things coincide. Coinciding again is LCM; largest equal grouping is HCF.
- Counting the starting instant. "How many more times" excludes the start.
Quick revision
- HCF: largest number dividing all; take the lowest power of each common prime. LCM: smallest number all divide into; take the highest power of every prime.
- Division: HCF by successive division (last non-zero divisor); LCM by the common-division ladder (product of divisors).
- For two numbers only: HCF times LCM equals the product.
- Fractions: HCF equals HCF(numerators) over LCM(denominators); LCM equals LCM(numerators) over HCF(denominators).
- HCF for largest equal parts (tiling, measuring). LCM for coincidence (bells, blinking) and smallest shareable quantity.
Practice
- Find the HCF and LCM of 24, 36 and 60.
- The HCF of two numbers is 12 and their LCM is 720. If one number is 48, what is the other?
- Find the greatest length that can measure 7 m, 3 m 85 cm and 12 m 95 cm an exact number of times.
- Find the least number of soldiers that can be arranged in rows of 12, 15 or 18 with none left over.
- Find the HCF of 3/4, 6/7 and 9/8.
- Traffic lights at three crossings change every 48, 72 and 108 seconds. If they change together at 8:00 a.m., when do they next change together?
- Two numbers are in the ratio 3 : 4 and their LCM is 240. Find the two numbers.
- Find the least number which, when divided by 6, 9 or 15, leaves a remainder of 4 in each case.