Tricks to find HCF and LCM of numbers

HCF & LCM

In this post we will learn how to find HCF and LCM of numbers quickly using different methods and tricks.

What is HCF?

HCF is Highest Common Factor, it is also known as GCD (Greatest Common Divisor or GCM (Greatest Common Measure). If a number A divides another number B exactly, We say that A is factor of B.
HCF or GCD or GCM of two or more number is the greatest number that divides each of them.

For Example: Take two numbers 16 & 40.
Factors of 16 = 1, 2,4, 8, 16
Factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40
Highest common factor is 8. So GCD of 16 & 40 is 8.

What is LCM?

LCM is Least Common Multiple. If a number A divides another number B exactly, we say that B is a multiple of A. LCM of two or more number is the least number which is exactly divisible by each of them.

For Example: Take two numbers 8 & 12.
Multiples of 8 = 8, 16, 24, 32, 40,..
Multiples of 12 = 12, 24, 36, 48,...
Least common multiple is 24. So LCM of 8 & 12 is 24.

Methods to find HCF

 1.  Prime Factorisation method:

• Resolve the given numbers into prime factors.

• Find the common prime factors of all the given numbers.

• The product of common prime factors with least powers gives HCF.

Example: Find the HCF of 24, 30 & 42.
Solution:  Step 1:  Expressing given numbers into prime factors.
           ⇒ 24 = 2 x 2 x 2 x 3
                   30 = 2 x 5 x 3
                   42 = 2 x 7 x 3
 Step 2:  Finding the common prime factors

2 & 3 are common factors of all the given numbers.



Step 3:  Product of common factors will give HCF ⇒  HCF = 2 x 3 = 6

 2.  Division Method:

• Divide the larger no. by smaller.

• Divide the divisor by remainder.

• Repeat second step till the remainder becomes zero. The last divisor is required HCF.

Example: Find the HCF of 26 & 455.
Solution:  Step 1:  455/26 = 13 remainder.
 Step 2:  26/13 = 0 remainder
 Step 3:  13 is the last divisor and required HCF.

NOTE: In case of more than two numbers, first find the HCF of first two numbers and then find HCF of this HCF with third number and so on.


 3.  Subtraction Method:
Remember this principle for this method: Any number which divides each of the two number also divides their sum, difference and sum & difference of any multiples of that numbers.

Example: Find the HCF of 42 & 70.
Solution: HCF of 42 and 70 = HCF of 42 and 28 (=70 – 42)
                ⇒ HCF of 28 and 14 (= 42 – 28)
                ⇒ HCF of 14 and 14 (= 28 – 14)
Therefore, HCF of 42 and 70 = 14.


 4.  For small numbers:

• HCF can’t be larger than the smallest number among the given numbers. So take the smallest number.

• Divide the other numbers by this number mentally. If it doesn’t divide all numbers then take highest factor of this number.

• Now divide all numbers with this factor, if it divides then it is the required HCF.

Example: Find the HCF of 8, 20, 28 & 44.
Solution:  Step 1:  Smallest number here is 8.
 Step 2:  8 doesn’t divide 28, so take its highest factor which is 4.
 Step 3:  4 divides all the other numbers, so our HCF is 4.


 5.  HCF of Fractions:
HCF = [HCF of numerator/LCM of denominator]

Example: Find the HCF of 7/2 & 35/4
Solution: HCF of 7 & 35 = 7
                       LCM of 2 & 4 = 4
             ⇒ HCF of fraction = 7/4

Methods to find LCM

 1.  Prime Factorization method:

• Resolve the given numbers into prime factors.

• Find the product of highest powers of all factors that occur in given number. This product is the required LCM.

Example: Find the LCM of 8, 12 & 15
Solution: Factors of given numbers:
                        8 = 2 x 2 x 2 = 23
                      12 = 2 x 2 x 3 = 22 x 3
                      15 = 3 x 5
       ⇒ LCM = 23 x 3 x 5 = 120


 2.  Division Method:

• Write down numbers separated by comma.

• Divide by any prime number which exactly divides at least any two of the given numbers.

• Set down the quotients and divided numbers in a line below the first.

• Repeat the process until you get a line of numbers which are prime to one another.

• Product of all divisors will be required LCM.                  

Example: Find the LCM of 8, 12 & 15.
Solution:

LCM = 2 x 2 x 2 x 3 x 5 x 4 x 3 = 1440  


 3.  LCM of Fractions:
LCM = [LCM of numerator/HCF of denominator]

Example: Find the LCM of 9/2, 3 & 21/2.
Solution: LCM of 9, 3 and 21 = 63
                   HCF of 2, 1 and 2 = 1
             ⇒ LCM of fraction = 63/1 = 63

Points to Remember

 1.  Product of two numbers = HCF x LCM

 2.  Co-prime numbers are those whose HCF is 1.

 3.  HCF is a factor of the given numbers and LCM of the given numbers.

 4.  If one of the factor of any number is ‘a’ and HCF is ‘x’ then number can be written as ‘ax’.

 5.  Greatest number which divides the number x, y & z leaving remainder a, b & c respectively = HCF of (x-a), (y-b), (z-c)

 6.  Greatest number which divides x, y and z leaving the same remainder in each case is = HCF of (x-y), (y-z), (z-x)

 7.  On dividing a number by a, b & c if we get (a-k), (b-k), (c-k) as remainder, then number will be = n x LCM of [a, b, c] - k

 8.  On dividing a number by a, b and c if we get k as remainder, then that number will be = n x LCM of [a, b, c] + k

 9.  If a number after adding k is exactly divisible by a, b and c then that number will be = n x LCM of [a, b, c] – k