Lowest Common Multiple definition
The lowest common multiple, often shortened to LCM, is the smallest positive multiple shared by two or more numbers. A multiple is the result of multiplying a number by a whole number. For example, multiples of 4 include 4, 8, 12, 16 and 20. Multiples of 6 include 6, 12, 18, 24 and 30. The first shared multiple is 12, so the lowest common multiple of 4 and 6 is 12.
LCM is different from HCF. The highest common factor is the largest factor shared by numbers, while the lowest common multiple is the smallest shared multiple. Factors divide into a number. Multiples are made by multiplying a number. This difference is important because GCSE questions often ask students to choose between HCF and LCM depending on whether the problem is about sharing into groups or matching repeated cycles.
Lowest common multiple is used in number, fractions, ratio, time, sequences, divisibility, prime factorisation and problem solving. At GCSE level, students use LCM to find common denominators, add or subtract fractions, solve repeated event problems and compare cycles. It can also appear in real-life contexts such as buses arriving, lights flashing, machines repeating or people meeting at regular intervals.
One simple method is to list multiples. To find the LCM of 8 and 12, list multiples of 8 as 8, 16, 24, 32 and 40. List multiples of 12 as 12, 24, 36 and 48. The first common multiple is 24, so the LCM is 24. This method is clear and works well for small numbers, but it can become slow when numbers are large.
Another method uses prime factorisation. For example, 18 can be written as 2 times 3 times 3. The number 24 can be written as 2 times 2 times 2 times 3. To find the LCM, include enough prime factors to build both numbers. This gives 2 times 2 times 2 times 3 times 3, which equals 72. Prime factorisation gives a systematic way to find LCM for larger numbers.
LCM is especially useful with fractions. To add 1 over 6 and 1 over 8, a common denominator is needed. The lowest common multiple of 6 and 8 is 24, so both fractions can be rewritten with denominator 24. This gives 4 over 24 plus 3 over 24, which equals 7 over 24. Using the LCM keeps the numbers as small as possible.
LCM also appears in cycle problems. If one bell rings every 6 minutes and another rings every 8 minutes, the bells will ring together every 24 minutes because 24 is the LCM of 6 and 8. This type of question shows why multiples are useful for repeated events. The LCM gives the first time when the cycles match again.
Related ideas include multiples, common multiples, factors, highest common factor, prime factorisation, common denominators, equivalent fractions and divisibility. A clear definition of lowest common multiple helps students understand that LCM is about the first shared result in repeated counting patterns. In exams, check whether the question asks for grouping and sharing, which often points to HCF, or repeated matching, which often points to LCM.