# Number System Tutorial II: Divisibility, Remainder, HCF & LCM

## LCM and HCF

LCM (Least common multiple) of a given set of quantities is the least possible quantity which can be divisible by each of the given quantities in the set.

HCF (Highest common factor)/ GCD (Greatest common divisor) of a given set of quantities is the highest possible quantity which can exactly divide each of the given quantities in the set.

### Prime factorization method for finding LCM and HCF

Prime factorization is an easy method for LCM and HCF for a given group of numbers. Let's look at an example to see how to find LCM and HCF of given set of numbers using prime factorization method.
Illustrative example: Find the LCM and HCF of 96, 108 and 120?
In this method, first find the prime factors of the given numbers. Prime factorized expressions of the numbers are given below.
96 = 2^5 xx 3
108 = 2^2 xx 3^3
120 = 2^3 xx 3 xx 5
LCM = product of maximum exponent forms of each distinct bases which are occurred in any of the given numbers.
:. "LCM" = 2^5 xx 3^3 xx 5 = 4320
HCF = product of maximum exponent form of distinct bases which are occurred commonly in all the numbers.
:. "HCF" = 2^2 xx 3 = 12

Find the LCM and HCF of 48, 36 and 72.?
Prime factorized expressions of the numbers are given below.
48 = 2^4 xx 3
36 = 2^2 xx 3^2
72 = 2^3 xx 3^2
LCM = product of maximum exponent forms of each distinct bases which are occurred in any of the given numbers.
:. "LCM" = 2^4 xx 3^2 = 144
HCF = product of maximum exponent form of distinct bases which are occurred commonly in all the numbers.
:. "HCF" = 2^2 xx 3 = 12

### LCM and HCF of fractions

"LCM of fractions" = "LCM of numerators"/"HCF of denominators"
Find the LCM of 6/5, 9/10 and 27/20?
LCM of 6/5, 9/10 and 27/20 = "LCM(6, 9, 27)"/"HCF(5, 10, 20)" = 54/4
"HCF of fractions" = "HCF of numerators"/"LCM of denominators"
Find the HCF of 16/15, 20/25 and 36/35?
HCF of 16/15, 20/25 and 36/35 = "HCF(16, 20, 36)"/"LCM(15, 25, 35)" = 4/525

### Key points on LCM and HCF

• Product of any two numbers is equal to the product of their LCM and HCF.
Eg: Let us consider two numbers 12 and 8.
LCM (12, 8) = 24
HCF (12, 8) = 4
12 xx 8 = 24 xx 4
i.e. product of 12 and 8 = LCM(12, 8) xx HCF (12, 8)
• HCF of any two numbers is a factor of the difference between the numbers.
Eg: Let A = 36 and B = 63
B - A = 63 - 36 = 27
HCF (36, 63) = 9
9 is a factor of 27.
• Let A and B are two numbers and their LCM and HCF are 'L' and 'H' respectively. A and B can be expressed in terms of their HCF in the following manner.
A = Hp and B = Hq, where p and q are two relatively prime numbers.
LCM of A and B can be expressed in the form; LCM (A, B) = Hpq

Eg: Let A = 48 and B = 32.
LCM = 96 and HCF = 16
A = 16 x 3, B = 16 x 2 : Here 3 and 2 are relatively prime numbers.
:. "HCF" = "16 and LCM" = 16 xx 2 xx 3 = 96
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