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L.C.M

The L.C.M of two or more numbers is the smallest number into which each of the numbers is the numbers will divide without a remainder.
Consider the numbers 6,12 and 8.
The multiples of 6, 12,18,24,30,36,42,48,54
The multiples of 12 are 12,24,36,60
THe multiples of 8 are 8,16,24,32,40,48,56
The L.C.M is 48.

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There are two methods to find L.C.M

(i) Prime factorization :
(a) Find prime factors for all given numbers.

(b) Take product of all prime factors having greatest exponents.

eg . 6 = 2 x 3
12 = 2² x 3
8 = 2³
L.C.M = 2³ x 3 = 24

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(ii) Common Division Method :

In this method, take product of divisors and undivided numbers to get L.C.M
eg .

L.C.M = 2³ x 3 = 24
L.C.M of 6,8,and 12 is 24.

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When to use L.C.M

(i) Find the least number which when divided by a, b, and c leaves a remainder of x in each case
In this case required no = (L.C.M of a,b,c) + x

(ii) When events occur after a certain time interval, then find the time zone when they meet together .

(iii) Find the smallest number which when divided by a,b,c leaves remainder x,y,z respectively.
In this case required number will be = (LCM of a,b,c) - k
k = a-x = b-y = c-z