LCM and HCF Questions

LCM and HCF are two entirely different terms in mathematics. The full form of LCM is Lowest Common Multiple or Least common multiple whereas HCF stands for Highest Common Factor.

The H.C.F interprets the greatest or highest factor present in between given two or more sets of numbers whereas L.C.M. defines the least or lowest number which is exactly divisible by two or more sets of numbers (exactly divisible means that they do not leave any remainder behind). 

HCF of any given set of numbers can not be greater than any of them whereas LCM of a given set of numbers cannot be smaller than any of them.

H.C.F is also known as the greatest common factor (GCF) whereas LCM is also known as the Least Common Divisor.

Stepwise Procedure to Find the HCF and LCM

There are several methods one can use to find the HCF and LCM of a given set of numbers.

• Prime factorization method
• Division method

HCF by Prime Factorization method 

As we know that the factors of a number are exact divisors of that specific number. 

To find the HCF of a given set of numbers by prime factorization, 

• Firstly, List the prime factors of those numbers. 

• After listing the factors, evaluate the product of the prime factors that are common to each of the given numbers. 

LCM by Prime Factorization method

To calculate the LCM of any given number,

1. Firstly, List the prime factors of the numbers
2. Write these prime factors in their exponent form.
3. Now, evaluate the product of only those factors that have the highest powers among these sets of numbers.

Evaluating HCF and LCM by Division Method

HCF by Division Method

To find the HCF by division method, 

1. Firstly, divide the larger number by the smaller number and check the remainder.
2. Make the remainder you got in Step1 as the divisor and the divisor of the above step as the dividend and conduct the division again.
3. Continue the division procedure until the remainder becomes equal to 0.
4. The last divisor will be the HCF of the given set of numbers.

LCM by Division Method

To find the LCM by division method, 

1. Divide the numbers by the smallest prime numbers, such that the prime number must divide 1 of the given numbers at least. 
2. Note down the quotients of the divisible numbers right below the numbers in the next row and copy the other numbers as they are.
3. Now, for the next division step assess the above quotients as the new dividends. Repeat the procedure and write the quotient below the numbers. 
4. Repeat the steps and divide the new dividends till you get 1.
5. Multiply all the prime numbers noted on the left-hand side to obtain the LCM of the given numbers.

LCM and HCF Questions Solved Examples

Example: 

Evaluate HCF of 198 and 360 using the division method.

Solution: 

Firstly, on divide 60 by 198, the remainder is 162. Then, refer to 162 as the divisor and 198 as the dividend and conduct the division again. The remainder is 36. Take 36 as the divisor and 162 as the dividend and conduct the division again. Here the remainder is 18. Take 18 as the divisor and 36 as the dividend and conduct the division again. Finally, the remainder is 0.

The last divisor, 18, will be the HCF of 360 and 198.

Example:

Find the LCM of 30 and 60.

Solution: 

Let's evaluate the LCM of 30 and 60 using the prime factorization method.

Prime factorization of 30 = 2 × 3 × 5 and 60 = 2 × 2 × 3 × 5. Note these prime factors in their exponential form as, 30 = 2¹ × 3¹× 5¹ and 60 = 2² × 3¹× 5¹. Now, evaluate the product of only those factors that have the highest powers. Which will be, 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60

Therefore, LCM(30,6) = 60

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FAQs on Calculator forLCM and HCF Questions with Steps

• How do you find LCM and HCF aptitude?

To find HCF and LCM for fractions quickly the below formulas are used,

Least Common Multiple(LCM) = Least Common Multiple(LCM) of numerator/Highest Common Factor(HCF) of denominator

Highest Common Factor(HCF)= Highest Common Factor(HCF) of numerator/ Least Common Multiple(LCM)of denominator 

• What is the rule of HCF and LCM?

The rule of HCF and LCM says that the product taken from two numbers is equal to the product of the LCM and HCF of those two numbers.

• How do you solve HCF and LCM questions?

Let’s consider HCF and LCM of two numbers are 4 and 60 respectively, if one number is 12, then the second number can be found by, assuming it is X and using formulas,

Product of Numbers= Product of their HCF and LCM

12× X = 4×60

X=4×60/12

X= 20

• What is HCF and LCM example?

For two numbers 45 and 30, HCF is 15 because it is the largest number that can divide 45 and 30 completely. Similarly, LCM for 45 and 30 is 90, because it is the smallest number which can be divided by 45 and 30.