How to Find : HCF & LCM Tricks ?

Basic definitions



  • HCF (Highest Common Factor) is the largest number that divides two or more given numbers exactly without leaving any remainder. Example, HCF of (8, 12) = 4

  • LCM (Least Common Multiple) is the smallest number that is exactly divisible by all the given numbers. Example, LCM of (8, 12) = 24.


Methods to find HCF


(A) Prime Factorization Method


Steps:



  1. Write the prime factors of each number.

  2. Multiply the common prime factors with lowest powers.


Example:
Find HCF of 36 and 60
36 = 2² × 3²
60 = 2² × 3 × 5
Common factors = 2² × 3 = 12
HCF = 12


(B) Division Method (Euclid’s method)


Steps:



  1. Divide the larger number by the smaller number.

  2. Divide the divisor by the remainder.

  3. Continue until remainder = 0.

  4. The last divisor = HCF.


Example:
Find HCF of 60 and 36
60 ÷ 36 → remainder 24
36 ÷ 24 → remainder 12
24 ÷ 12 → remainder 0
HCF = 12


(C) Shortcut Trick (for 2 or 3 numbers)



  • If numbers are multiples, smaller = HCF.
    Example: HCF of 12, 24, 36 = 12

  • If numbers are consecutive, HCF = 1.
    Example: HCF of 8, 9, 10 = 1


Methods to find LCM


(A) Prime Factorization Method


Steps:



  1. Write the prime factors of each number.

  2. Multiply all prime factors with highest powers.


Example:
Find LCM of 12 and 18
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36


(B) Division Method (for 2 or more numbers)


Steps:



  1. Write numbers in a row.

  2. Divide by any prime that divides at least one number.

  3. Continue till all become 1.

  4. Multiply all divisors = LCM


Example:
Find LCM of 12, 18, 24







































Step

Numbers

                 Divisor

1

12, 18, 24

                 ÷2 → 6, 9, 12

2

6, 9, 12

                 ÷3 → 2, 3, 4

3

2, 3, 4

                 ÷2 → 1, 3, 2

4

1, 3, 2

                 ÷3 → 1, 1, 2

5

1, 1, 2

                  ÷2 → 1, 1, 1

LCM = 2 × 3 × 2 × 3 × 2 = 72

(C) Relation Between HCF and LCM


For any two numbers,


HCF × LCM = Product of numbers


Example:
If numbers are 12 and 18,
HCF = 6, so
LCM = (12 × 18) / 6 = 36


Short Tricks & Memory Tips









































Case

 Trick

   Example

When one number divides the other

      Smaller = HCF, Larger = LCM

(4, 8): HCF=4, LCM=8

For co-prime numbers

       HCF = 1, LCM = Product

(4, 9): HCF=1, LCM=36

For 3 consecutive numbers

       HCF = 1

(10,11,12): HCF=1

For 3 consecutive even numbers

       HCF = 2

(12,14,16): HCF=2

For fractions

      HCF = HCF of numerators / LCM of denominators


  
 

      LCM = LCM of numerators / HCF of  denominators


  

 


Example Mixed Problem


Find HCF and LCM of 16a²bc³, 32abc, 64a³bc²


LCM: Take highest powers
= 64a³bc³


HCF: Take lowest powers
= 16abc²


HCF = 16abc²
LCM = 64a³bc³


Example:
Find HCF of 36 and 60



  • 36 = 2² × 3²

  • 60 = 2² × 3 × 5
    Common = 2² × 3 = 12
    HCF = 12


Examples




































Numbers

       HCF

      LCM

8, 12

           4

        24

5, 7

           1

        35

9, 18

           9

        18

15, 25, 35

           5

        525

6, 8, 12

           2

        24

 
























Memory Table

Concept

     Use Lowest Power

   Use Highest Power

HCF

                  Yes

                No

LCM

                   No

               Yes