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Prime Factorization Of HCF And LCM

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Last updated date: 23rd May 2024
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What is HCF and LCM?



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:

  • Prime factorization method

  • Division method.


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.

FAQs on Prime Factorization Of HCF And LCM

1. What is the division method for finding the LCM and HCF?

In the division method for finding the LCM and HCF, there are four steps that involve mathematical operations which are divide, multiply, subtract, and bring down. The first step involves the division of the given number by the divisor by the identification of the suitable integers. Then in the next step, the divisor and integer are multiplied to get the number to be subtracted from the dividend. The subtracted number is brought down as the remainder and another digit from the dividend and this process is continued till a 0 remainder is received or a number that is less than the divisor.

2. What are some of the properties of HCF?

Some of the properties of HCF of two or more given numbers are listed below:

  • It divides, without any remainder, each of the numbers.

  • It is a factor of each of the numbers which are given.

  • It is always either less than or equal to one but cannot be greater.

  • If the numbers given are prime numbers, then the HCF is always equal to one.

3. What are some of the properties of LCM?

Some of the properties of LCM of two or more given numbers are listed below:

  • It can not be less than any of the numbers. For example, the LCM of 7 and 4 is equal to 28 and is not less than any of the numbers.

  • The LCM is the greater number itself when in the case of two given numbers one is the factor of the other. For example, the LCM of 2 and 4 is equal to 4 which is the greater number itself.

4. What are some of the differences between a factor and a multiple?

The number which can be divided for a certain number of times by another number leaving no remainder is known as the multiple of that number while the factor is the number which after being divided by a specific number leaves no remainder. For a given number, the factors can be finite while the multiples are infinite. Another key difference is that the resulting factor is always less than or equal to the given number while the resulting multiples are greater than or equal to the given number.