Prime Factorization for Least Common Multiple

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prime factorization

The least common multiple (LCM) of two or more numbers is the smallest common multiple, which is common to the numbers. For example, LCM of 2 and 4 is 4, because 2 times of 2 is 4, so 4 is common multiples for both 2 and 4. To find LCM for two or more numbers, we have three methods:

  • Listing the Multiples
  • Prime Factorisation Method
  • Division Method

Here we will discuss how to find using the prime factorization method.

LCM by Prime Factorisation

To discover the LCM of at least two numbers, we first discover all the prime factors of the given numbers and think of the one underneath the other. Take one factor from every normal gathering of variables and discover their item. Pair the factor with other ungrouped factors. The resultant is the LCM of given numbers. Here are the steps to find LCM by prime factorization.

  • Resolve each given number into its prime factors and express the variables obtained in the example structure.
  • Discover the result of the most noteworthy forces of the multitude of components that happen in any of the given numbers.
  • The item gotten in the second step is the necessary least basic difference (L.C.M).

With help of these steps we can easily calculate LCM of two or more numbers.

Examples

Let us see some examples of finding LCM using prime factorization.

  1. LCM of 12, 16 and 24

12 = 2 x 2 x 3

16 = 2 x 2 x 2 x 2

24 = 2 x 2 x 2 x 3

Now, write all the prime factors of 12, 16 and 24.

12 x 16 x 24 = 2 x 2 x 3 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Pair the factors.

LCM (12, 16, 24) = 2 x 2 x 2 x 2 x 3 = 48

  1. LCM of 9 and 15

9 = 3 x 3

15 = 3 x 5

Now, write all the prime factors of 9 and 15

9 x 15 = 3 x 3 x 3 x 5

As we can see here we can only pair for 3. Therefore,

LCM (9,15) = 3 x 3x 5 = 45

  1. LCM of 32, 48 and 72

32 = 2 x 2 x 2 x 2 x 2

48 = 2 x 2 x 2 x 2 x 3

72 = 2 x 2 x 2 x 3 x 3

Clubbing all the prime factors together we get;

32 x 48 x 72 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 x 2 x 2 x 2 x 3 x 3

Pairing the common factors, we get;

LCM of (32, 48, 72) = 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288

So, we can see how easy it is to find the lcm using prime factorization. In the same way we can determine lcm for other pairs of numbers as well.

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