# Lowest Common Multiple (LCM)

The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples. For example, the common multiples of 4 and 6 are 12, 24, 36, ... . The lowest of these is 12. So, the lowest common multiple of 4 and 6 is 12.

The LCM of the two numbers is the product of the prime factors counted the maximum number of times they occur in any of the numbers.

LCM of 12 and 18

The prime factorisation of 12 and 18 are:

12 = 2 × 2 × 3

18 = 2 × 3 × 3

In these prime factorisation, the maximum number of times the prime factor 2 occurs is two; this happens for 12. Similarly, the maximum number of times the factor 3 occurs is two; this happens for 18. Thus, in this case LCM = 2 × 2 × 3 × 3 = 36.

LCM of 24 and 90

The prime factorisation of 24 and 90 are:

24 = 2 × 2 × 2 × 3

90 = 2 × 3 × 3 × 5

In these prime factorisation, the maximum number of times the prime factor 2 occurs is three; this happens for 24. Similarly, the maximum number of times the prime factor 3 occurs is two; this happens for 90. The prime factor 5 occurs only once in 90.

Thus, LCM = (2 × 2 × 2) × (3 × 3) × 5 = 360

LCM of 40, 48 and 45

The prime factorisation of 40, 48 and 45 are:

40 = 2 × 2 × 2 × 5

48 = 2 × 2 × 2 × 2 × 3

45 = 3 × 3 × 5

The prime factor 2 appears maximum number of four times in the prime factorisation of 48, the prime factor 3 occurs maximum number of two times in the prime factorisation of 45, The prime factor 5 appears one time in the prime factorisation of 40 and 45, we take it only once.

Therefore, required LCM = (2 × 2 × 2 × 2) × (3 × 3) × 5 = 720

Example 1: In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?

The distance covered by each one of them is required to be the same as well as minimum. The required minimum distance each should walk would be the lowest common multiple of the measures of their steps.

Thus, find the LCM of 80, 85 and 90. The LCM of 80, 85 and 90 is 12240. The required minimum distance is 12240 cm.

Example 2: Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.

First, find the LCM of 12, 16, 24 and 36.

LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144

144 is the least number which when divided by the given numbers will leave remainder 0 in each case. But you need the least number that leaves remainder 7 in each case.

Therefore, the required number is 7 more than 144. The required least number = 144 + 7 = 151.