HCF / GCD and LCM - Definition, Formula, Full Form, Examples

The full form of HCF/GCD is the Highest Common Factor/Greatest Common Divisor (Both terms mean the same thing), while the full form of LCM is the Least Common Multiple. HCF is the largest number that divides two or more numbers without leaving a remainder, whereas LCM is the smallest multiple that is divisible by two or more numbers.

HCF is the Highest Common Factor, which can be calculated for two or more numbers. It is denoted by HCF(a, b), where "a" and "b" are the numbers for which we want to find the highest common factor.

LCM can be seen in two or more numbers. It is denoted by LCM(a, b), where "a" and "b" are the numbers for which we want to find the least common multiple.

HCF or GCD Definition

• The HCF or GCD of two numbers is defined as the largest number that can exactly divide both numbers.
• HCF is the  Highest Common Factor that divides all the given numbers exactly. Therefore, HCF is also known as the Greatest Common Divisor or GCD.

Example: Find the HCF of 6 and 18.
Solution:

Divisors of 6 = 1, 2, 3, 6
Divisors of 18 = 1, 2, 3, 6, 9, 18

HCF = greatest common divisor
HCF = 6

LCM Definition

The LCM of two or more numbers is defined as the smallest number that can be divided by all of the numbers. LCM is the least number that is a common multiple of all the given numbers.

Example: Find the LCM of 6 and 18.
Solution:

Multiple of 6 = 6, 12, 18, 24, 30, …
Multiple of 18 = 18, 36, 54, …
LCM = first common multiple (least common multiple)
LCM = 18

HCF / GCD and LCM Formula

To find the HCF and LCM formulas, let's assume that the numbers given are a and b. The relationship between HCF and LCM states that the product of a and b is equal to the product of HCF and LCM. 

(LCM of two numbers) × (HCF of two numbers) = Product of two numbers

Mathematically, this can be written as:

LCM(a, b) × HCF(a, b) = a × b

How to Find HCF and LCM?

These are the most famous methods used to calculate HCF and LCM :

• Division method
• Prime factorization method

Let's learn about all these methods in detail.

HCF by Division Method

The easiest way to understand how to find HCF by the Division Method is by going back to simple division.

The following are the steps for better understanding this method :

Step 1: Take the smaller number as the divisor and the larger number as a dividend.
Step 2: Perform division. If you get the remainder as 0, then the divisor is the HCF of the given numbers.
Step 3: If you get a remainder other than 0 then take the remainder as the new divisor and the previous divisor as the new dividend.
Step 4: Perform steps 2 and step 3 until you get the remainder as 0.

Example: Find out the HCF of 36 and 48.

Solution:

Using the division method for HCF

Hence, HCF = 12

LCM by Division Method

The following steps can be followed to find the Least Common Multiple by the Division Method:

Step 1: Check whether the given numbers are divisible by 2 or not.
Step 2: If the number is divisible by 2 then divide and again check for the same. If the numbers are not divisible by 2 then check 3, and so on.
Step 3: Perform step 2 until you get 1 in the end.

Example: Find out the LCM of 36 and 48.

Solution:

Using the division method for LCM

Hence, LCM = 2 × 2 × 2 × 2 × 3 × 3 = 144

HCF by Prime Factorization

Finding HCF by Prime Factorization can be done by following the given steps:

Step 1: Find out the prime factors of the given number.
Step 2: Check the occurrence of a particular factor. Find out the common factors and choose them in HCF.
Step 3: Multiply the occurrence of common factors. And this will be the HCF Of the given numbers.

Example: Find out the HCF of 18 and 90.

Solution:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 90 = 2 × 3 × 3 × 5

Now, HCF = 2 × 3 × 3 = 18

LCM by Prime Factorization 

Finding LCM by Prime Factorization is done by following the given steps:

Step 1: Find out the prime factors of the given number.

Step 2: Check the occurrence of a particular factor. If a particular factor has occurred multiple times in the given number, then choose the maximum occurrence of the factor in LCM. It can also be found out by checking the powers of the factors. The factor having greater power will be chosen between the numbers.

Step 3: Multiply all the maximum occurrences of a particular factor. And this will be the LCM Of the given numbers.

Example: Find out the LCM of 18 and 90.

Solution:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 90 = 2 × 3 × 3 × 5

Now, LCM = 2 × 3 × 3 × 5 = 90

Alternate method:

Prime factors of 18 = 2 × 3 × 3
Prime factors of 18 = 2¹ × 3²

Prime factors of 90 = 2 × 3 × 3 × 5
Prime factors of 90 = 2¹ × 3² × 5¹

Chosen factors for LCM = 2¹ × 3² × 5¹

Therefore, LCM = 2 × 9 × 5 = 90.

HCF vs LCM

Here are some key differences between HCF and LCM:

Related Articles:

• How to Find HCF
• How to Find GCD of more than 2 Number?
• Finding GCD by Prime Factorization
• Properties of GCD

Solved Question on HCF / GCD and LCM

Question 1: Find out the LCM and HCF of 18, 30, and 90 by prime factorization.

Solution: 

Prime factors of 18 = 2 × 3 × 3
Prime factors of 30 = 2 × 3 × 5
Prime factors of 90 = 2 × 3 × 3 × 5

LCM: 2 × 3 × 3 × 5 = 90
HCF: 2 × 3 = 6

Question 2: Find out the LCM and HCF of 318 and 504.

Solution: 

Prime factors of 318 = 2 × 3 × 53
Prime factors of 504 = 2 × 2 × 2 × 3 × 3 × 7

LCM: 2 × 2 × 2 × 3 × 3 × 7 × 53 = 26712
HCF: 2 × 3 = 6

Question 3: Find out the HCF of 24 and 36.
Solution:

Let's find out the HCF of 24 and 36 by division method, 

Therefore,

HCF = 2 × 2 × 3 = 12

Question 4: Find out the LCM of 24 and 36.

Solution:

Let's find out the LCM of 24 and 36 by division method, 

Therefore,

LCM = 2 × 2 × 3 × 2 × 3 = 72

Question 4: Find out the LCM and HCF of 15 and 70. Also, verify the relationship between LCM, HCF, and given numbers.

Solution:

Prime factors of 15 = 3 × 5
Prime factors of 70 = 2 × 5 × 7

LCM: 2 × 3 × 5 × 7
HCF: 5

Verifying the relationship:

LCM × HCF = 2 × 3 × 5 × 5 × 7 = 1050
Product of two numbers = 15 × 70 = 1050

From above you can see that,
LCM (15, 70) ×  HCF(15, 70) = Product of two numbers

Hence Verified.

Practice Questions on HCF ( or GCD ) and LCM

Question 1: Find the HCF of 36 and 60.

Question 2: What is the LCM of 12, 18, and 24?

Question 3: Two numbers have an HCF of 8 and an LCM of 96. If one of the numbers is 32, find the other number.

Question 4: Calculate the HCF and LCM of 45 and 75.

Question 5: The product of two numbers is 2400, and their HCF is 20. Find their LCM.

Question 6: Find the HCF of 72, 108, and 144.

Question 7: Two cyclists are riding on circular tracks. One completes a round in 12 minutes, and the other in 18 minutes. After how many minutes will both cyclists meet at the starting point if they start together? (Hint: Find the LCM of their times.)

Question 8: Three friends have ropes of lengths 24 meters, 36 meters, and 48 meters. They want to cut their ropes into equally smaller pieces without any leftovers. What is the maximum possible length of each smaller piece they can cut? (Hint: Find the HCF of the rope lengths.)

Suggested Quiz

10 Questions

The LCM of two numbers is 30, and their HCF is 15. If one of the numbers is 30, what is the other?

• A

  30

• B

  10

• C

  15

• D

  20

Explanation:

Say another number =y
Product of two numbers = Product of HCF and LCM
y ✕ 30 = 15 ✕ 30, so y=15 is the required answer.

Three numbers are in the ratio of 2 : 3 : 4 and their H.C.F. is 20. Their L.C.M. is:

• A

  80

• B

  180

• C

  240

• D

  480

Explanation:

Let the numbers be 2x, 3x and 4x
HCF =20
⇒The HCF of 2x, 3x, and 4x is given as 20. This means that: x=20.
So, the actual numbers are 40, 60, and 80.
The LCM of the three numbers is 240.
so, 240 is the required answer.

What is the least number which when divided by 4, 5, 6 and 7 leaves a remainder 3, but when divided by 9 leaves no remainder?

• A

  1683

• B

  417

• C

  843

• D

  423

Explanation:

LCM of 4,5,6,7 is 420
we know that common multiple of 4,5,6,7 is in the form of 420y (where y is any natural number)
(420y+3) should be divisible by 9 
if y =1, 423/9, remainder=0
When 423 is divided by 4, 5, 6, 7, it gives remainder 3 and when divided by 9 leaves a remainder 0.
So, the least number which when divided by 4, 5, 6 and 7 leaves a remainder 3, but when divided by 9 leaves no remainder is 423

A, B and C start at the same time in the same direction to run around a circular track. A completes a round in 252 seconds, B in 308 seconds and C in 198 seconds, all starting at the same point. When will they meet again at the starting point?

• A

  26 minutes 18 seconds

• B

  42 minutes 36 seconds

• C

  45 minutes

• D

  46 minutes 12 seconds

Explanation:

LCM of 252, 308 and 198 = 2772.
Therefore, they will meet at the starting point again after 2772 seconds = 46 minutes and 12 seconds.

The least number which when divided by 5, 6, 7 and 8 leaves a remainder of 3, but when divided by 9 leaves no remainder, is:

• A

  1677

• B

  1683

• C

  2523

• D

  3363

Explanation:

LCM of 5, 6, 7 and 8 = 840.
Therefore, required number is of the form 840y + 3.
Least value of y for which 840y + 3 is divisible by 9 is 2.
For y = 2, 840y + 3 = 1683.

Calculate the HCF of 1.08, 0.36 and 0.9.

• A
  0.03
• B
  0.9
• C
  0.18
• D
  0.108

Explanation:

Let's rewrite the numbers as 108/100, 36/100 and 90/100. Now, HCF of 108, 36 and 90 is 18. Therefore, 18/100 = 0.18 is our answer.

Three numbers are in the ratio 1:2:3 and their HCF is 12. The numbers are:

• A
  4, 8, 12
• B
  5, 10, 15
• C
  10, 20, 30
• D
  12, 24, 36

Explanation:

Given ratios are 1:2:3 Let the three numbers be x, 2x and 3x, when we find HCF of x, 2x and 3x we get x because 1, 2, 3 are primes. Also given HCF is 12

x = 12 

So other numbers are 2x = 24 and 3x = 36.

The sum of two numbers is 528 and their HCF is 33. The number of pairs of numbers satisfying the above condition is:

• A

  4

• B

  6

• C

  3

• D

  5

Explanation:

Let the required numbers be 33x and 33y.
Then, 33x + 33y = 528
33(x+y)=528.
But, 528 = 2 ✕ 2 ✕ 2 ✕ 2 ✕ 3 ✕ 11.
33(x+y)= 528= 16 ✕ 33
So, x + y = 16
Co-primes with the sum 16 are: (1, 15), (3, 13), (5, 11) and (7, 9).
Hence, 4 is the required answer.

The product of two number is 4107. If the HCF of these numbers is 37, then the greater number is:

• A

  101

• B

  107

• C

  111

• D

  185

Explanation:

Let the required numbers be 37x and 37y.
Then, 37x * 37y = 4107
So, x * y = 4107/(37*37) = 3.
Co-primes with product 3 are (1,3).
Therefore, the greater number = 3 * 37 = 111.

Express 1095/1168 in its simplest form.

• A

  13/16

• B

  15/16

• C

  17/26

• D

  25/26

Explanation:

To simplify the fraction, we need to divide both the numerator (1095) and the denominator (1168) by their HCF
To find HCF:

• Divide 1168 by 1095:1168÷1095=1 remainder 73.
  (Since 1168 is larger than 1095, dividing gives a quotient of 1 and a remainder of 73.)
• Now divide 1095 by 73:1095÷73=15 remainder 0.
  (Since the remainder is now 0, the process stops, and 73 is the HCF of 1095 and 1168.)

Simplify the Fraction by Dividing Both Numbers by Their HCF
1168/1095 ​= (1168÷73)/(1095÷73​) = 16/15​
Thus, the simplified fraction is: 15/16= 1095/1168.

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