Formula Sheets for General Aptitude (Part A): HCF & LCM , Download PDF!

Formula Sheet On HCF & LCM

HCF and LCM

Highest Common Factor (HCF): The highest common factor of two or more numbers is the greatest common divisor, which divides each of those numbers an exact number of times. The process to find the HCF is:
a. Express the numbers given as a product of prime numbers separately i.e. find factors of numbers.
b. Take the product of prime numbers common to all the given numbers.

Example 1: Find HCF of 540 and 1024.
Solution:
Step 1: Express the numbers given as a product of prime numbers.
540 = 2 × 2 × 3 × 3 × 3 × 5 = 22 × 33 × 51
1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 210
Step 2: Take the product of prime numbers common to all the given numbers.
We can see that only 22 is common in both the given numbers. Thus, H.C.F. = 2 × 2 = 4

Example 2: Find the HCF of 27, 81, 165, 360.
Solution:
Step 1:
27 = 3 × 3 × 3 = 33
81 = 3 × 3 × 3 × 3 = 34
165 = 3 × 5 × 11
360 = 23 × 32 × 5
Step 2:
We can see that only 3 is common to all the given numbers. Thus, HCF = 3.
Sometimes finding HCF becomes very calculative and time-consuming as the given numbers can be many and their respective values are also large. In that case,
Property: Let us suppose two numbers N1 and N2 are given. So,
HCF of (N1, N2) = HCF of difference between N1, N2 or any factor of the difference between the given numbers.

Example 3: Find the HCF of 36 and 54.
Solution:
By ordinary method: 36 = 22 × 32 and 54 = 21 × 33. Thus, HCF = 21 × 32 = 18
Or the Difference between 36 and 54 = 18
Check whether 18 divides both 36 and 54 or not. Here, 18 divides 36 and 54 completely.
Thus, the HCF of 36 and 54 will be 18. So, both the cases have the same answer.
Example 4: Find the HCF of 210, 360 and 540.
Solution:
HCF of (210, 360 and 540) = HCF (360 – 210 and 540 – 360) = HCF (150 and 180) = 30
Check whether 30 divides all the three numbers or not. Here, 30 divides 210, 360 and 540 completely. Thus, HCF of 210, 360 and 540 will be 30.
Note: Try to find the difference between given numbers which is as minimum as possible.

Example 5: Find the HCF of 2190, 1800, 1890 and 2520.
Solution:
Here, Finding the HCF of these four numbers will be hectic as these numbers are large and their factorization will be time consuming. So, try to find shortest possible difference between the given numbers. Here, shortest possible difference will be between 1890 and 1800 which is 90.
Now, Check whether 90 divides all the four numbers or not. Here, 90 divides 1800, 1890 and 2520 completely but not 2190. So, HCF will not be 90 but a factor of 90.
Factors of 90 = 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90. Out of all the twelve factors of 90, highest factor that divides all the given numbers is 30. Thus, HCF will be 30.

Least Common Multiple (LCM): The least common multiple (LCM) of two or more numbers is the smallest of the numbers, which is exactly divisible by each of them, e.g. consider two numbers 18 and 24.
The multiples of 18 are: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, ....
The multiples of 24 are: 24, 48, 72, 96, 120, 144, 168, 192, 216, ......
The common multiples of both 18 and 24 are 72, 144, 216, ....
The least common multiple is 72.
Here again, try to break the words in reverse order and understand the concept. Firstly, find the multiples of the numbers. Secondly, the common multiples of the numbers and finally the least out of those will be the LCM. The process to find the LCM is:
a. Express the numbers given as a product of prime numbers separately i.e. find factors of numbers
b. Take the product of prime factors of the given numbers after eliminating repetition of the common factors.

Example 6: Find LCM of 36 and 54.
Solution:
Step 1: Express the numbers given as a product of prime numbers.
36 = 22 × 32
54 = 21 × 33
Step 2: Take the product of prime factors of the given numbers after eliminating repetition of the common factors.
Here, eliminating the common factors 21 and 32 and multiplying the remaining factors i.e.
22 and 33.
So, LCM = 22 × 33 = 108

Example 7: Find the LCM of 210, 360 and 540.
Solution:
Step 1: Express the numbers given as a product of prime numbers.
210 = 2 × 3 × 5 × 7
360 = 23 × 32 × 5
540 = 22 × 33 × 5
Step 2: Take the product of prime factors of the given numbers after eliminating repetition of the common factors.
Here, eliminating the common factors 21, 31 and 51 and multiplying the remaining factors
i.e. 23 and 33, 5 and 7.
So, LCM = 23 × 33 × 5 × 7 = 7560

Relationship between HCF and LCM:
The relationship between any two numbers x and y and their HCF and LCM: x × y = LCM × HCF
Proof: Let us take any two numbers such as 14 and 78.
Factorization of 14 = 2 × 7 and 78 = 2 × 3 × 13
Product of 14 and 78 = 14 × 78 = 1092
HCF of 14 and 78 = 2
LCM of 14 and 78 = 2 × 3 × 7 × 13 = 546
Product of HCF and LCM = 2 × 546 = 1092
Thus, Product of 14 and 78 = HCF × LCM of 14 and 78.

A special case of finding HCF: There are some cases when HCF is asked in question but two numbers (N1 and N2) are not given instead their sum and LCM is given. So, in that case:
HCF of (N1 and N2) = HCF of (Sum of N1 and N2, LCM of N1 and N2)
Proof: Find the HCF of 36 and 54.
36 = 22 × 32
54 = 21 × 33
HCF = 21 × 32 = 18
Here, the Sum of 36 and 54 = 90
And Sum of 36 and 54 = 22 × 33 = 108
HCF of (Sum of 36 and 54, LCM of 36 and 54) = HCF of (90 and 108) = 18
So, in both cases HCF is the same.
HCF of fraction values: To calculate the HCF of fraction values, we calculate the ratio of
HCF of all the numerators to LCM of all the denominators.