Overview: HCF and LCM of Numbers | Quantitative Techniques for CLAT PDF Download

HCF

HCF stands for highest common factor. It is also known as greatest common divisor or GCD. Let us consider a₁ and a₂ as the two natural numbers. If these two natural numbers a₁ and a₂ are divisible by exactly the same number which is n then ‘n’ is called the common factor of a₁ and a₂. The highest number of all these common factors of a₁ and a₂ is called as the HCF or GCD. For example, the highest common factor of 18 and 24 is 6.
To find HCF there are two methods:

• Factorization method
• Division method

(i) Factorization Method: In this method, write the numbers in the standard form. The prime numbers that are common to all the numbers and their factors will be our required HCF.
Examples 1: Find the HCF of 160, 220, 340.
Sol: First, we will start by writing the numbers in the standard form. Thus, the numbers will be written as:
160 = 2³ x 2² x 5
220 = 2² x 5 x 11
340 = 2² x 5 x 17
So, the numbers common in the above sequence are 2² and 5. Thus, the HCF of the given numbers will be 2² x 5 = 20.

(ii) Division Method: For this method, take two of the given numbers, divide the greater by the smaller and then divide the divisor by the reminder. Now again divide the divisor of this division by the next remainder found and repeat this method until the remainder is zero. The last divisor that is found will be the HCF of the two numbers asked. If there are three numbers given and you need to find the HCF of three numbers then find the HCF of this two numbers and the third number.

Example 2: What will be the HCF of 327 and 436?
Sol: Here first we have to check which number is smaller. And we will use that small number as a divisor to divide the larger number. Here 327 is a smaller number and 436 is a larger number. So we will divide 436 by 327.
Here the remainder is 109. Now we will divide 327 by 109.
So, the HCF of 327 and 436 is 109.

Example 3: Find the HCF of 324, 576, and 784
Sol: The division method becomes a bit complicated when you try and find the HCF of three numbers. We will start by taking two numbers and then we will find HCF of the third number. Let’s start with 784 and 576,
576) 784 (1
-576
208) 576 (2
– 416
160) 208 (1
– 160
48) 160 (2
– 144
16) 48 (3
– 48
0
So, the HCF of 576 and 784 is 16.
Now find the HCF of 16 and the remaining number i.e. 324.
16) 324 (20
– 320
4) 160 (40
– 160
0
Thus, we have found that the HCF of 576, 784, and 324 is 4.

LCM

LCM stands for least common multiple. Suppose that there are two natural numbers, n₁ and n₂. The smallest natural number ‘p’ that is exactly divisible by n₁ and n₂ is known as the LCM of n₁ and n₂. For example, 15 is the LCM of 3 and 5.
To solve LCM of numbers there are two methods. They are

• Factorization method
• Division method

(i) Factorization Method: In this method, just like HCF, you have to write the numbers in the standard form. Then the product of prime numbers that appears at least once in any of these numbers raised to the highest available power is called the LCM of these numbers. We will understand more clearly with the help of an example.

Example1 : Find the LCM of 160, 220, and 340.
Sol: As given above first we need to write the numbers in a standard form. They will be following:
160 = 2³ x 2² x 5
220 = 2² x 5 x 11
340 = 2² x 5 x 17
Now you need to find the numbers that appear at least once in all the numbers. So, the LCM of above numbers will be 2² x 2³ x 5 x 11 x 17 = 29,920.

(ii) Division Method: In division method, find the prime number that is at least one time common to all the numbers. Write all these given numbers in a line, divide them by the prime numbers that are selected and write the quotient below the numbers. If there is no prime number which divides the selected number. Write the same number below it. Don’t stop unless you get the quotient which is prime to each other. LCM will be the product of all the divisors and the prime numbers in the last line of the numbers.

Example 2: Find the LCM of 12,18, and 27
Sol: Here three numbers are given and we will start by dividing these numbers by the smallest possible divisor.
2 |12, 18, 27
3 |6, 9, 27
3 | 2, 3, 9
2, 1, 3
Thus, LCM will be 2 x 3 x 3 x 2 x 1 x 3 = 108
Note: The product of HCF and LCM of two numbers equals the product of these two numbers.

Example 3: The LCM and HCF of two numbers are 156 and 4 respectively. If one number is 52 find the other number.
Sol: 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 x 4 = 52 × x
⇒ x = 12
So, the required answer is 12.