[Maths Class Notes] on Prime Factorization Of HCF And LCM Pdf for Exam

The largest number that divides each of the supplied numbers without leaving any remainder is the highest common factor (HCF) of two or more given numbers. The lowest of the common multiples of two or more numbers is called the least common multiple (LCM). Learning HCF and LCM in mathematics is important since it helps us in solving day-to-day problems involving grouping and sharing.

For example, the HCF of 24 and 36 is 12, as 12 is the greatest number that can divide both the numbers without leaving any remainder. In the same way, the LCM of two or more numbers is the lowest number that is a common multiple of the given numbers.

What are the Methods to Find HCF and LCM?

There are a number of methods to find the HCF and LCM of given numbers. But most common methods are:

Using Prime Factorization Methods to Find HCF and LCM

A factor is a number that divides a number exactly into a different number. For example, 5 divides 35 into 7, hence 5 is a factor of 35, while 7 is also a factor of 35. One is a factor of every number. The number itself is also a factor of any number.

Factorization is the method of writing a number as a product of several of its factors. Factorization is not considered meaningful as compared to division, but it finds its use when we wish to find the simplest constituents of a number and to represent a number as a product of the same.

Example 1: Find the HCF of 50 and 75 by the prime factorization method.

Ans: Prime factors of 50 = 2 × 5 × 5

Prime factors of 75 = 3 × 5 × 5

If we analyse factors of both numbers, the common factors are 5 × 5.

Therefore, the HCF of (50, 75) is 25.

Example 2: Find HCF of 390, 702, and 468 by using the prime factorization method.

Ans: Prime factors of 390 = 1 × 2 × 3× 5 × 13. 

Prime factors of 702 = 1 × 2 × 3 × 3 × 3 × 13. 

Prime factors of 468 = 1 × 2 × 2 × 3 × 3 × 13. 

Here, the common factors of 390, 702, and 468 are 1, 2, 3, 13. 

Therefore, HCF of 390, 702, and 468 is 2 × 3 × 13 = 78.

LCM by Prime Factorization

Prime factorization is one of the methods to find the LCM of two or more numbers. We can similarly separately find the prime factors of any numbers that are given and then find the LCM by identifying the common factors among them.

To calculate the least common multiple of given numbers, we follow the steps given below:

List all the prime factors of the given numbers and note down the common prime factors.

The LCM of the numbers is the calculated by-product of the common prime factors as well as the uncommon prime factors of the numbers.

Example 1: What is the L.C.M of 36 and 14 by using the prime factorization method?

Ans: To find the LCM, we have to multiply all prime factors. But note that the common factors are included only once.

The prime factors of 36 = 2 × 2 × 3 × 3 = 2² × 3².

The prime factors of 14 = 2 × 7.

L.C.M = 2² × 3² × 7 = 2 × 2 × 3 × 3 × 7 = 252.

Therefore, the least common multiple (L.C.M) of 36 and 14 = 252.

Example 2: What is the L.C.M of 5, 4, and 16 by using the prime factorization method?

Ans: To find the LCM, we have to multiply all prime factors. But note that the common factors are included only once.

The prime factors 5 = 5 × 1.

The prime factors 4 = 2 × 2.

The prime factors 16 = 2 × 2 × 2 × 2 = 2⁴.

L.C.M = 2⁴ × 5 = 2 × 2 × 2 × 2 × 5 = 80.

Therefore, the least common multiple (L.C.M) of 5, 4, and 16 = 80.

Conclusion

The highest common factor between two or more numbers is known as the HCF. The “Greatest Common Divisor” is another name for it. Similarly, the lowest number that is a common multiple of the given numbers is the least common multiple (LCM) of those numbers.