Finding L.C.M and H.C.F of two numbers and applications

L.C.M can be defined as Least Common Multiple of all the numbers.
H.C.F can be defined as Highest Common Factor of all the numbers.

Important formulae:

1. LCM*HCF=product of the numbers.
2. If any number is factor of other then LCM is bigger one and HCF is the other.

In general to find LCM,we need to write all the multiples of given numbers and find common number in the series similarly to find HCF of few given numbers,we need to find all factors of given numbers and identify big such factor common for both the numbers but this is a time taking process so we are here for you with some other trick.

First let us know about the traditional methods to find LCM,HCF.
In traditional method,to find LCM of 4,15

1. we have to write multiple series of both the numbers.
2. Identify common number in both.

Multiples of 4: 4,8,12,16,20,24,28,32,36,40,45,58,52,56,60,64.........

Multiples of 15: 15,30,45,60,75..........

we can say 60 is LCM of 4,15 but this takes more time.

In traditional method to find HCF to two numbers(20,60)

1. We have to write series of all factors for the given numbers.
2. Identify the common factors and note the highest such number.

Factors of 60: 1,2,3,4,5,6,12,15,20,30,60

Factors of 20:1,2,4,5,10,20

we can say 20 is the Highest Common Factor for both 20,60.

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But the methods used above take more time so here is the shortcut:(before learning this, go through prime factorization)

Algorithm for finding LCM:

1. Prime Factorize the given numbers.(It will be in form num1: a^x1*b^y1*c^z1..... and num2: a^x2*b^y2*^z2.....)
2. Select every prime factor along with the power(greater of x1 and x2...)
3. Now simplify the expression.

Concept of Negative numbers with LCM:

Problems related to finding remainders can be solved using this concept.

Ex: A number when divided by 5,6,7 leaves the remainder 1,2,3 respectively, find the number.

Solution: difference is (5-1)=(6-2)=(7-3)

so LCM(5,6,7)-difference (4)

is the answer.

LCM and HCF solved problems:

Example: Finding LCM of 20,60

Solution:

step 1: prime factorize the numbers

20=4*5=(2^2)*(5^1)

60=3*4*5=(2^2)*(3^1)*(5^1)

step 2: select every prime number with greatest power;

   2: greater(2,2)=2^2

   3:greater(0,1)=3^1

   5:greater(1,1)=5^1

step 3:simplify expression

 2^2*3^1*5^1=4*5*3=60.

Algorithm for finding HCF:

1. Prime Factorize the given numbers.(It will be in form num1: a^x1*b^y1*c^z1..... and num2: a^x2*b^y2*^z2.....)
2. Select every prime factor along with the power(lower of x1 and x2...)
3. Now simplify the expression.

Example: Finding HCF of 20,60

Solution:

step 1: prime factorize the numbers

20=4*5=(2^2)*(5^1)

60=3*4*5=(2^2)*(3^1)*(5^1)

step 2: select every prime number with the lowest power;

   2: lower(2,2)=2^2

   3:lower(0,1)=3^0

   5:lower(1,1)=5^1

step 3:simplify expression

 2^2*3^0*5^1=4*5=20.

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LCM Calculator tool : 

Number 1: Number 2:

Hope you enjoyed the trick,comment if you have any queries.