What is multiple ?
A multiple is the result of multiplying a
number by an integer (not a
fraction).
1)
Multiples of 2 are 2,4,6,8,10,12,14,16,18,20,22,24,26,…
Pictorial representation :
Pictorial representation :
Example: 10 is a multiple of 2,i.e; 10=2×5
10 is a multiple of 5,i.e;10=5×2
11 is not a multiple of 2 or 5.
Other Definition:
We get a multiple of a
number when we multiply it by another number. Such as multiplying
by 1, 2, 3, 4, 5, etc, but not zero. Just like the multiplication
table.
The multiples of 4 are
:4,8,12,16,20,24,28,32,36,40
The multiples of 5 are
:5,10,15,20,25,30,35,40,45,50.
What is Common multiples ?
Suppose we have listed first some
multiple of two numbers.Then the multiples found in both the list of multiples
of two numbers are the common multiples.
Example:
The multiples of 4 are :4,8,12,16,20,24,28,32,36,40(44,48,52,56,60,…and
so on)
The multiples of 5 are :5,10,15,20,25,30,35,40,45,50(55,60,… and so
on
The Common multiples of the given list
of multiples of numbers 4 and 5 are 20 and 40…(60,80 ,…and so on)
What is Lowest Common Multiple ?
Lowest Common multiple is the smallest or least of all the common multiples of numbers.
Example:Find the LCM of 3 and 6.
Step1 :First we have to find the multiples of 3 and
6.
Step 2: From the above,We can easily find the common
multiples of 3 and 6.These are 6,12,18,… and so on.
Step 3:From the common multiples ,We have to find
the smallest or least common factor from all the common factors of number 3 and
6.Here the least common factor is 6.
Methods to Find the LCM
There are mainly these two methods of finding the
LCM of numbers.
1:Prime Factorisation Method
2:Common Division Method
1:Prime Factorisation Method:By Using Prime Factorisation
Method,Find the LCM of 18,30 and 50.
Step 1:First find the prime factors of all the
numbers
|
Prime
Factors of 18
|
Prime
Factors of 30
|
Prime
Factors of 50
|
Prime Factors of 18 : 2,3,3
Prime Factors of 30 : 2,3,5
Prime Factors of 50 : 2,5,5
Here 2 comes maximum 1 times ,3 comes maximum
2 times and 5 comes maximum 2 times.
So the LCM =2×3×3×5×5=450
2.Common Division Method:
By Using the Common Division Method,Find the LCM of
72,108 and 144
Step 1: Divide by the smallest prime number that can
divide at least one of the numbers.Being the ones that can not be divided down
as it is.
Step 2:Keep dividing by the smallest possible prime
number till the last row has only prime numbers or numbers that have only 1 as
a common factor(co-prime).
Step 3:Multiply all the common factors and all the
numbers in the last row to get the LCM.
Pictorial Representation:
|
2
|
72,108,144
|
|
2
|
36,
54, 72
|
|
2
|
18, 27, 18
|
|
2
|
9, 27,
18
|
|
3
|
9, 27,
9
|
|
3
|
3, 9, 3
|
|
|
1, 3, 1
|
Then the LCM =2×2×2×2×3×3×3=432
Using LCM:
Find the smallest number which when divided
by 21,45 and 36 will leave a remainder in each case.
In this case First we have to find the
LCM of all these numbers and then add 7 with the LCM.
Pictorial Representation:
|
2
|
21, 45, 36
|
|
2
|
21, 45, 18
|
|
3
|
21, 45, 9
|
|
3
|
7, 15, 3
|
|
|
7, 5, 1
|
Then the LCM =2×2×3×3×7×5=1260
If we divide 1260 by 21, 45 and 36 we
will not get the remainder. So to get the remainder of 7 , we need to add 7 to
the LCM.
Then the new LCM=1260+7=1267
Then the smallest number when divided by
21, 36 and 45 that will leave a remainder of 7 is 1267.
Relation Between HCF and LCM
The Product of the two numbers is equal
to product of the HCF and LCM.
Example: Find the HCF and LCM of 15 and 9 and check
the above .
Solution: HCF of 15 and 9 is 3
LCM of 15 and 9 is 45
Then the product of HCF and LCM=3×45=135
Then the product of the two numbers =15×9=135
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