LCM (Lowest Common Multiple)
LCM (Lowest Common Multiple) of given two or more than two numbers is the lowest number which is exactly divisible by the given numbers.
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Among the common multiples of the given
numbers, the lowest one is the LCM.
For example, 60 is the lowest
number which is exactly divisible by 10, 12 and 15. So 60 is the LCM of 10, 12
and 15.
Here,
The multiples of 10, M10
= {10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, …}
The multiples of 12, M12
= {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The multiples of 15, M15
= {15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, …}
Common multiples = {60, 120, …}
The lowest common multiple = 60
∴ LCM
of 10, 12 and 15 is = 60
LCM
can be calculated by using the following methods:
a.
Set
of Multiple Method
b.
Prime
Factorization Method
c.
Common
Division Method
Let's find the LCM of 12, 16 and 24
by three different methods.
Set of Multiples Method
The multiples of 12, M12
= {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, …}
The multiples of 16, M16
= {16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, …}
The multiples of 24, M24
= {24, 48, 72, 96, 120, 144, 168, 192, 216, 240, …}
Common multiples = {48, 96, 144,
…}
The lowest common multiple = 48
∴ LCM
= 48
Prime Factorization Method
To find LCM by prime
factorization method, we should apply the following steps:
Step
1: Find prime factors of
the given numbers and list them.
Step
2: Write the common
factors and the remaining factors.
Step
3: Find the product of
these factors.
Here,
∴ 12 =
2 × 2 × 3
16
= 2 × 2 × 2 × 2
24
= 2 × 2 × 2 × 3
The product of the common and the remaining factors is the LCM.
∴ LCM = 2 × 2 × 2 × 3 × 2 = 48
Common Division Method
To calculate LCM by common
division method, we should apply the following steps:
Step
1: Arrange the given
numbers in a row and separate by commas.
Step
2: Divide all numbers by
the smallest prime numbers.
Step
3: Repeat the process
till the numbers left in the last row is 1.
Step
4: Multiply all the prime
divisors to get LCM.
Let’s find the LCM of 12, 16 and
24 by common division method.
∴ LCM
= 2 × 2 × 2 × 2 × 3 = 48
Worked Out Examples
Example
1: Find the LCM of 30, 40 and
60 by the set of multiples method.
Solution:
Here,
Multiples
of 30, M30 = {30, 60, 90, 120, 150, 180, 210, 240, …}
Multiples
of 40, M40 = {40, 80, 120, 160, 200, 240, 280, …}
Multiples
of 60, M60 = {60, 120, 180, 240, 300, 360, …}
Common
multiples = {120, 240, …}
Lowest
common multiple is the LCM,
∴ LCM = 120 Ans.
Example
2: Find the LCM of 50, 60 and
80 by the prime factorization method.
Solution:
Here,
The
prime factors of 50, 60 and 80 are calculated as follows:
∴ 50 = 2 × 5 × 5
60 = 2 × 2 × 3 × 5
80 = 2 × 2 × 2 × 2
× 5
∴ LCM = Common factors × remaining factors
= 2 × 2 × 5 × 5 × 3 × 2
= 1200 Ans.
Example
3: Find the LCM of 30, 40 and
80 by using the common division method.
Solution:
Here,
Example
4: Find the least number which
is exactly divisible by 12, 40 and 60 without leaving a remainder.
Solution:
Here,
The required least number is the LCM of 12, 40, and 60.
Example
5: Find the least number when
divided by 40, 60 and 80 leaves a remainder 6 in each case.
Solution:
Here,
∴ The required number = LCM + remainder
= 240 + 6
= 246 Ans.
If you have any question or
problems regarding the LCM, you can ask here, in the comment section below.
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