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Parin Sir > Quantitative Aptitude (Maths) > H. C. F. & L. C. M. OF NUMBERS > To find L. C. M. of Given Numbers > To find L. C. M. of Given Numbers

To find L. C. M. of Given Numbers

June 1, 2020
Category: To find L. C. M. of Given Numbers

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☼ To find L. C. M. of Given Numbers :

■ L. C. M. by Definition : Write all the multiples of each number and take common multiples of all multiples and then select the least multiple among them. This least common multiple is called the L. C. M.

e.g. Find the L. C. M. of 12 and 18.

Soln. The multiples of given numbers are as follows

12 $\large \rightarrow$ 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, . . .

18 $\large \rightarrow$ 18, 36, 54, 72, 90, 108, 126, 144, 162, . . .

the common multiples of given numbers are :

36, 72, 108, 144, . . .

And among above, the least common multiple is 36.

i.e. L. C. M. = 36. Hence, [12, 18] = 36.

■ L.C.M. By Factorization in the form of power : Resolve each one of the given numbers into prime factors. Then, the product of highest powers of all the factors, gives the L.C.M.

e.g. Find the L.C.M. of 96, 108, 280.

Soln. 96 = $2^{5}\times 3$ , 108 = $2^{2}\times 3^{3}$ and 280 = $2^{3} \times 5\times 7$ .

$\therefore$ L.C.M. = 2⁵ $\times$ 3³ $\times$ 5 $\times$ 7 = 30240

■ L.C.M. By Factorization in the form of multiplication of primes : Resolve each one of the given numbers into prime factors. Then, the product of highest powers of all the factors, gives the L.C.M.

e.g. Find L.C.M. of 99000, 220500 and 2102100.

Soln. 99000 = 2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 5 $\times$ 5 $\times$ 5 $\times$ 11

220500 = 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 5 $\times$ 5 $\times$ 5 $\times$ 7 $\times$ 7

2102100 = 2 $\times$ 2 $\times$ 3 $\times$ 5 $\times$ 5 $\times$ 7 $\times$ 7 $\times$ 11 $\times$ 13

Taking the common, pair wise common & remaining factors, we get :

L.C.M. = 2 $\times$ 2 $\times$ 2 $\times$ 3 $\times$ 3 $\times$ 5 $\times$ 5 $\times$ 5 $\times$ 7 $\times$ 7 $\times$ 11 $\times$ 13

= 63063000

Hence, [99000, 220500, 2102100] = 63063000.

■ L.C.M. By Common Division Method (Short cut Method) : Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number expect 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

e.g. Find the L.C.M. of 48, 24, 36 and 54.

Solution: