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Playing With Numbers

HCF and LCM

You must have played hide and seek a number of times. In this game, you have to find your hidden friends. Finding factors and multiples is like playing hide and seek with numbers. You have the number, yet you have to seek the hidden factors and multiples through the concept of HCF and LCM. The chapter below shall make the game of hiding and seek with numbers easy.

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Highest Common Factor (HCF)

We already know that a number which exactly divides two or more given numbers is the common factor of the given numbers. In factorization, we intend to find the factors of a given number. Now what to do if we had to find the common factor between two or more numbers. This is where the concept of HCF comes into play.

The HCF or Highest Common Factor of two or more numbers is the greatest common factor of the given set of numbers. In other words, HCF is the greatest number which exactly divides two or more given numbers.

HCF and LCM Source: pixabay

HCF by Listing Method

The listing method involves the process of listing the factors of the given numbers. For example, find the HCF of 20 and 35.

  • All possible factors of 20 are 1,2,4,5,10 and 20
  • All possible factors of 60 are 1,3,4,5,6,10,12,15,20,30,60

The common factors of the given numbers are : 1,2,4,5,10,20. The greatest among all other numbers is 20, so it shall be the HCF of both the numbers.

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HCF by Prime Factorization

Before finding HCF by prime factorization we need to know the concept of the same. Let’s take a number say, 45. Now the factors of 45 are 1,3,5,9,15 and 45 itself. Now, apart from 3 and 5 the other numbers 9 and 15 are composite numbers. We hence further factorize them with 9= 3×3 and 15=3×5.

So the factors of 45 shall be only 1,3,3, and 5. This is prime factorization. We now define prime factorization as the process of expressing the number as the product of its prime factors. The prime factors include only prime numbers and not composite numbers.

When we find HCF by prime factorization method, we are finding the greatest common factor among the prime factors or numbers. Steps to be followed for the method are:

  1. Find the prime factors of each of the given number.
  2. Next, we identify the common prime factors of the given numbers
  3. We then multiply the common factors. The product of these common factors is the HCF of the given numbers.

Let us use these steps in the example below: find the HCF of  36 and 48.

Step 1: Finding prime factors individually:

HCF and LCM

  • All possible factors of 36 are: 2×2×3×3×1
  • All possible factors of 48 are: 2×2×2×2×3×1

Step 2: Choose out the common factors: 2×2×3

Step 3: Multiply all the common factors to get the HCF of the given numbers:

Here the given numbers are 36 and 48. The product of the common factors: 2×2×3 = 12. So the HCF for the numbers 36 and 48 is 12.

Lowest Common Multiple (LCM)

The LCM of a set of two or more numbers is the smallest of their common multiples. Multiples mean the numbers which follow as the result of multiplying the number with numbers like 1,2,3 etc. To find the common multiples all we need to do is see what numbers end up to be the common multiple for all the given numbers.

For example, when we find the LCM of 9 and 12 we need to find the common multiples. The common multiples of 9 are 36,72,108 etc… The smallest of these is 36, hence 36 shall be the LCM of 9 and 12.

LCM by Prime factorization

To find the LCM using prime factorization method we need to follow the below-mentioned steps:

  • Find the prime factors of numbers individually.
  • From all the factors, identify the maximum number of times each prime factor appears.
  • The product of the prime factors occurring in maximum numbers is the LCM of the given set of numbers.

Let us use the steps in the following example: find the LCM of 8 and 24.

Step 1: First find the prime factors of the numbers 8 and 24

HCF and LCM

  • Prime factors of 8 = 2×2×2
  • Prime Factors of 24= 2×2×2×3

Step 2: Choose out the number occurring a maximum number of times. The number 2 occurs 3 times and 3 occurs 1 time. number occurring the maximum number of times is 2×2×2×3.

Step 3: The product of these numbers is 24. So the LCM of 8 and 24 is 24.

LCM by Division Method

For calculating the LCM by division method we need to follow the below mentioned steps:

  • First, write all the given numbers in a single row but separated by commas.
  • Find the least prime number that divides at least two numbers from the set of given numbers.
  • Write the quotients exactly below the respective number. The numbers which are not divisible by that prime number have to be written as they are, below the respective number.
  •  Keep repeating the step 2 till no two numbers are divisible by the same number.
  • To find the LCM, multiply the divisors and remaining quotients. The product of all is the LCM of the given set of numbers.

Let us use the above steps in the example given below: find the LCM of 15,30,90.

Step 1: Rewrite the numbers in a row separated by commas.

HCF and LCM

Step 2,3 and 4: Find the prime factors, the number with no quotients is written as it is. The step is repeated until we reach the stage where no prime factors are possible to find.

HCF and LCM

Step 5: Now we find the product of divisors and the remaining quotient. For the given numbers 15, 30 and 90 the product divisors and quotients are 2×3×5×3 = 90.

From the above set of rules and examples, we can calculate the HCF and LCM for any set of numbers. Finding the HCF and LCM using these steps will help your task of finding the same easy and fast.

Solved Question for You on HCF and LCM

Question 1: The LCM and HCF of two numbers are 156 and 4 respectively. If one number is 52 find the other number.

Answer : As it is given that the product of two numbers is equal to the product of HCF and LCM of two numbers. And HCF and LCM are given to us. Also, one of the numbers is given to us. Thus, we need to find the other number.

So, LCM x HCF = product of two numbers
=> 156 × 4 = 52 × x
=> x = 12

So, the required answer is 12.

Question 2: How can one calculate the HCF?

Answer: One can find the highest common factor by carrying out the multiplication of all the factors which appear in both lists. Therefore, the HCF of 60 and 72 is 2 × 2 × 3 which turns out to be 12.

Question 3: What is the use of LCM?

Answer: LCM can be used before the addition, subtraction, or comparison of fractions can take place. The LCM of more than two integers also happens to be well-defined: it is the smallest positive integer whose division can take place by each of them.

Question 4: Who is credited with developing the method for finding HCF?

Answer: Euclid is credited with developing the method for finding HCF.

Question 5: Tell about the formula of finding the HCF?

Answer: Greatest Common Measure(GCM) and Greatest Common Divisor(GCD) are the terms that an individual can use it to refer to HCF. For example, HCF of 60 and 75 = 15 as 15 happens to be the highest number which can divide both 60 and 75 in an exact manner. One can find HCF by making use of the prime factorization method or by using the division method.

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