To find Least Common Multiple by using Prime Factorization Method

To find least common multiple by using prime factorization method is discussed here.

To find the LCM of two or more numbers, we first find all the prime factors of the given numbers and write them one below the other. Take one factor from each common group of factors and find their product. Multiply the product with other ungrouped factors. The resultant is the LCM of given numbers.

Step I:

Resolve each given number into its prime factors and express the factors obtained in exponent form.

Step I:

Find the product of the highest powers of all the factors that occur in any of the given numbers.

Step III:

The product obtained in Step II is the required least common multiple (L.C.M).

For example:

1. Find the least common multiple (L.C.M) of 9 and 15 by using prime factorization method.

Solution:

Step I:

Resolving each given number into its prime factors.

9 = 3 × 3 = 3².

15 = 3 × 5.

Step II:

The product of all the factors with highest powers.

= 3^2 × 5 = 3 × 3 × 5 = 45.

Step III:

The required least common multiple (L.C.M) of 9 and 15 = 45.

2. What is the least common multiple (L.C.M) of 16 and 28 by using prime factorization method?

Solution:

Step I:

Resolving each given number into its prime factors.

16 = 2 × 2 × 2 × 2 = 2⁴.

28 = 2 × 2 × 7 = 2² × 7.

Step II:

The product of all the factors with highest powers.

= 2⁴ × 7 = 2 × 2 × 2 × 2 ×7 = 112.

Step III:

The required least common multiple (L.C.M) of 16 and 28 = 112.