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# Least Common Multiple (LCM)

Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the numbers.

Example: LCM of 3 and 4 = 12 because 12 is the smallest number which is a multiple of both 3 and 4 (In other words, 12 is the smallest number which is divisible by both 3 and 4).

We can find LCM using prime factorization method or division method.

# How to find out LCM using prime factorization method

Step 1 : Express each number as a product of prime factors.

Step 2 : LCM = The product of highest powers of all prime factors.

Example 1: Find out LCM of 8 and 14

Step 1 : Express each number as a product of prime factors. (Reference: Prime Factorization)

8 = 23

14 = 2 × 7

Step 2 : LCM = The product of highest powers of all prime factors.

Here the prime factors are 2 and 7

The highest power of 2 here = 23

The highest power of 7 here = 7

Hence LCM = 23 × 7 = 56 Example 2: Find out LCM of 18, 24, 9, 36 and 90

Step 1 : Express each number as a product of prime factors.

18 = 2 × 32

24 = 23 × 3

9 = 32

36 = 23 × 32

90 = 2 × 5 × 32

Step 2 : LCM = The product of highest powers of all prime factors.

Here the prime factors are 2, 3 and 5

The highest power of 2 here = 23

The highest power of 3 here = 32

The highest power of 5 here = 5

Hence LCM = 23 × 32 × 5 = 360

# How to find out LCM using division Method (shortcut method)

Step 1: Write the given numbers in a horizontal line separated by commas.

Step 2: Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.

Step 3: Write the quotients and undivided numbers in a line below the first.

Step 4: Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.

Step 5: LCM = The product of all the divisors and the numbers in the last line.

Example 1: Find out LCM of 8 and 14
 2 8,14 4,7

Hence Least common multiple (L.C.M) of 8 and 14
= 2 × 4 × 7
= 56 Example 2: Find out LCM of 18, 24, 9, 36 and 90

 2 18,24,9,36,90 2 9,12,9,18,45 3 9,6,9,9,45 3 3,2,3,3,15 1,2,1,1,5

Hence Least common multiple (L.C.M) of 18, 24, 9, 36 and 90
= 2 × 2 × 3 × 3 × 2 × 5
= 360