LCM and HCF: The LCM stands for Least Common Multiple, whereas the full form of HCF is Highest Common Factor. LCM is the smallest multiple that can be divided by two or more integers. The highest common factor (HCF) is the greatest number that divides two or more numbers with no remainder.
LCM Definition:
The least common multiple of two or more numbers is the smallest number that can be divided by all of them. LCM may happen in two or more numbers. It is represented by LCM(a, b), where “a” and “b” are the quantities for which we seek the least common multiple.
For example, let’s calculate the LCM of 4 and 16.
Solution:
Multiple of 4 = 4,8,12,16 …
Multiple of 16 = 16, 32, 52, …
LCM = first common multiple (least common multiple)
LCM = 16
HCF Definition:
HCF stands for Highest Common Factor, which can be noticed in two or more numbers. It is represented by HCF(a, b), where “a” and “b” are the numbers for which we seek the highest common factor. HCF is also known as the Greatest Common Divisor, or GCD. The highest common factor (HCF) of two numbers is the greatest number that can divide them exactly. HCF is the greatest common factor that splits all of the numbers provided perfectly.
For example, let us calculate the HCF of 4 and 16
Solution:
Divisors of 4 = 1, 2, 4,
Divisors of 16 = 1, 2, 4, 8,
HCF = greatest common divisor
HCF = 4
HCF and LCM Formula:
If the numbers “a” and “b” are given in order to find the HCF and LCM formulas. The relation between HCF and LCM states that the product of a and b equals to the product of HCF and LCM.
Product of 2 numbers = (LCM of two numbers) × (HCF of two numbers)
This could be written as : a x b = LCM(a, b) × HCF(a, b)
Methods to Find LCM and HCF?
Here are the most popular ways for calculating LCM and HCF:
 Division method
 Prime factorization method
LCM using Division Method:
To find using the division approach, we divide the following procedures can be taken to find the Least Common Division using Division Method:
Step 1: Verify whether the given numbers are divisible by 2.
Step 2: If the number is divisible by 2, divided it and verify for the same. If numbers are not divisible by 2, then try 3, and so on.
Step 3: Continue step 2 till you get 1.
For example, find the LCM of 9 and 21.
Solution:
LCM = 3 × 3 × 7 = 63
HCF through Division Method:
The fastest way to understand how to calculate HCF by Division Method is to go back to simple division.
Here are the steps to help you understand this method:
Step 1: Use the smaller number as a divisor and the larger number as the dividend.
Step 2: Perform division. If the residual is 0, then the divisor is the HCF of the provided numbers.
Step 3: If you get a nonzero as remainder, use it as new divisor and use the previous divisor as the dividend.
Step 4: Repeat steps 2 and 3 until the remainder equals 0.
For example, find the HCF of 9 and 21.
Solution:
LCM through Prime Factorization:
To find LCM using Prime Factorization, follow the steps below:
Step 1: Identify the prime factors of the given integer.
Step 2: Analyze the occurrence of a specific factor. If a specific factor appears more than once in the provided number, select the factor with the highest incidence in LCM. It can also be determined by comparing the powers of the components. The component with the greatest power will be picked from among the numbers.
Step 3: Multiply all of the highest occurrences of a specific factor. And this will be the LCM of the provided numbers.
For example, find the LCM of 9 and 21.
Solution:
Prime factors of 9 = 3 × 3
Prime factors of 9 = 3^{2}
Prime factors of 21 = 3 x 7
Prime factors of 21 = 3^{1} × 7^{1}
Chosen factors for LCM = 3^{2} × 7^{1}
Therefore, LCM = 9 × 7 = 63.
HCF using prime factorization:
To find HCF using Prime Factorization, follow the steps below:
Step 1: Get the prime factors of the given integer.
Step 2: Identify the occurrence of a specific factor. Find the common factors and choose them in HCF.
Step 3: Multiply the number of common factors. And this will be the HCF for the provided numbers.
For example, find the HCF for 18 and 90.
Prime factors for 9 = 3 x 3
Prime factors for 21 = 3 x 7
Now, HCF = 3.
Practice Quizzes
LCM And HCF Quiz 1 – Coming Soon 
LCM And HCF Quiz 2 – Coming Soon  LCM And HCF Quiz 3 – Coming Soon 
LCM and HCF Examples Problems.
Q1. The H.C.F. and L.C.M. of two numbers are 50 and 250 respectively. If the first number is divided by 2, the quotient is 50. The second number is
(A) 50
(B) 100
(C) 105
(D) 125
Ans – D
First number = (50×2)=100
Second number = [(50×250) /100 ] =125
Q2. The number of prime factors in the expression (6)^{10} x (7)^{17} x (11)^{27} is
(A) 44
(B) 54
(C) 64
(D) 71
Ans: C
Since 2, 3, 7, 11 are prime numbers and the given expression is
2^{10} × 3^{10} × 7^{17} × 11^{27}.
The number of prime factors in the given expression is
(10 + 10 + 17 + 27) = 64.
Q3. When janaki made necklaces of either 16 beads, 20 beads or 36 beads, not a single bead was left over. What could be the least number of beads janaki had?
(A) 700
(B) 710
(C) 720
(D) 740
Ans: C
Required number of beads = L.C.M. of 16, 20, 36 = 720.
Q4. Find the least number exactly divisible by 12, 15, 20 and 27.
(A) 340
(B) 540
(C) 440
(D) 320
Ans : B
Required number = L.C.M of 12, 15, 20, 27.
∴ L.C.M = 3 × 4 × 5 × 9 = 540.
Hence, required number is 540.
The option B is correct.
Q5. Find the greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively.
(A) 112
(B) 117
(C) 127
(D) 137
Ans : C
Required number = H.C.F of (1657 – 6) and (2037 – 5) = H.C.F of 1651 and 2032.
∴ Required number = 127.
Hence, option C is correct.
Q6. Find the HCF and LCM Of 2/3; 8/9; 16/81; and 10/27.
(A) 3/80
(B) 80/3
(C) 81/2
(D) None
Ans : B
Q7. The H.C.F of two number is 11 and their L.C.M is 693. If one of the numbers is 77, find the other.
B) 37
C) 47
D) 54
Ans : B
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Frequently Asked Questions about HCF and LCM
Q1. What is the formula for LCM and HCF?
The product of two numbers is equal to the product of their HCF and LCM.
As a result, HCF of two numbers = product of two numbers/L.C.M of two numbers.
The LCM of two numbers is the product of two numbers divided by their H.C.F.
Q2. How do we find the LCM and HCF?
We can find LCM and HCF using the prime factorization and long division methods.
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