# Least Common Multiple

The least common multiple (L.C.M.) of two or more numbers is the smallest number which can be exactly divided by each of the given number.

Let us find the L.C.M. of 2, 3 and 4.

Multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, ...... etc.

Multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ...... etc.

Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, ...... etc.

Common multiples of 2, 3 and 4 are 12, 24, 36, ...... etc.

Therefore, the smallest common multiple or least common multiples of 2, 3 and 4 is 12.

Consider the following.

(i) 12 is the least common multiple (L.C.M) of 3 and 4.

(ii) 6 is the least common multiple (L.C.M) of 2, 3 and 6.

(iii) 10 is the least common multiple (L.C.M) of 2 and 5.

We can also find the L.C.M. of given numbers by their complete factorisation.

To find for instance, L.C.M. of 24, 36 and 40, we first factorise them completely.

24 = 2 × 2 × 2 × 3 = 2$$^{3}$$ × 3$$^{1}$$

36 = 2 × 2 × 3 × 3 = 2$$^{2}$$ × 3$$^{2}$$

40 = 2 × 2 × 2 × 5 = 2$$^{3}$$ × 5$$^{1}$$

L.C.M. is the product of highest power of primes present in the factors.

Therefore, L.C.M. of 24, 36 and 40 = 2$$^{3}$$ × 3$$^{2}$$ × 5$$^{1}$$ = 8 × 9 × 5 = 360

Solved examples to find the lowest common multiple or the least common multiple:

Find the L.C.M. of 8, 12, 16, 24 and 36

8 = 2 × 2 × 2 = 2$$^{3}$$

12 = 2 × 2 × 3 = 2$$^{2}$$ × 3$$^{1}$$

16 = 2 × 2 × 2 × 2 = 2$$^{4}$$

24 = 2 × 2 × 2 × 3 = 2$$^{3}$$ × 3$$^{1}$$

36 = 2 × 2 × 3 × 3 = 2$$^{2}$$ × 3$$^{2}$$

Therefore, L.C.M. of 8, 12, 16, 24 and 36 = 2$$^{4}$$ × 3$$^{2}$$ = 144

To Find Lowest Common Multiple by using Division Method

Relationship between H.C.F. and L.C.M.

Worksheet on H.C.F. and L.C.M.

Word problems on H.C.F. and L.C.M.

Worksheet on word problems on H.C.F. and L.C.M.

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