# 21, 28 and 32 LCM LCM of 21, 28 and 32 is equal to 672. The comprehensive work provides more insight of how to find what is the lcm of 21, 28 and 32 by using prime factors and special division methods, and the example use case of mathematics and real world problems.

what is the lcm of 21, 28 and 32?
lcm (21   28   32) = (?)
21 => 3 x 7
28 => 2 x 2 x 7
32 => 2 x 2 x 2 x 2 x 2

= 2 x 2 x 7 x 3 x 2 x 2 x 2
= 672
lcm (21, 28 and 32) = 672
672 is the lcm of 21, 28 and 32.

where,
21 is a positive integer,
28 is a positive integer,
672 is the lcm of 21, 28 and 32,
{2, 2, 7} in {3 x 7, 2 x 2 x 7, 2 x 2 x 2 x 2 x 2} are the most repeated factors of 21, 28 and 32,
{3, 2, 2, 2} in {3 x 7, 2 x 2 x 7, 2 x 2 x 2 x 2 x 2} are the the other remaining factors of 21, 28 and 32.

Use in Mathematics: LCM of 21, 28 and 32
The below are some of the mathematical applications where lcm of 21, 28 and 32 can be used:

1. to find the least number which is exactly divisible by 21, 28 and 32.
2. to find the common denominators for the fractions having 21, 28 and 32 as denominators in the unlike fractions addition or subtraction.
Use in Real-world Problems: 21, 28 and 32 lcm
In the context of lcm real world problems, the lcm of 21, 28 and 32 helps to find the exact time when three similar and recurring with different time schedule happens together at the same time. For example, the real world problems involve lcm in situations to find at what time all the bells A, B and C toll together, if bell A tolls at 21 seconds, B tolls at 28 seconds and C tolls at 32 seconds repeatedly. The answer is that all bells A, B and C toll together at 672 seconds for the first time, at 1344 seconds for the second time, at 2016 seconds for the third time and so on.

Important Notes: 21, 28 and 32 lcm
The below are the important notes to be remembered while solving the lcm of 21, 28 and 32:
1. The repeated and non-repeated prime factors of 21, 28 and 32 should be multiplied to find the least common multiple of 21, 28 and 32, when solving lcm by using prime factors method.
2. The results of lcm of 21, 28 and 32 is identical even if we change the order of given numbers in the lcm calculation, it means the order of given numbers in the lcm calculation doesn't affect the results.
For values other than 21, 28 and 32, use this below tool:

## How-to: What is the LCM of 21, 28 and 32?

The below solved example with step by step work shows how to find what is the lcm of 21, 28 and 32 by using either prime factors method and special division method.

Solved example using prime factors method:
What is the LCM of 21, 28 and 32?

step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 21
B = 28
C = 32

What to be found:
find the lcm of 21, 28 and 32

step 2 Find the prime factors of 21, 28 and 32:
Prime factors of 21 = 3 x 7
Prime factors of 28 = 2 x 2 x 7
Prime factors of 32 = 2 x 2 x 2 x 2 x 2

step 3 Identify the repeated and non-repeated prime factors of 21, 28 and 32:
{2, 2, 7} are the most repeated factors and {3, 2, 2, 2} are the non-repeated factors of 21, 28 and 32.

step 4 Find the product of repeated and non-repeated prime factors of 21, 28 and 32:
= 2 x 2 x 7 x 3 x 2 x 2 x 2
= 672
lcm(20 and 30) = 672

Hence,
lcm of 21, 28 and 32 is 672

Solved example using special division method:
This special division method is the easiest way to understand the entire calculation of what is the lcm of 21, 28 and 32.

step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 21, 28 and 32

What to be found:
lcm (21, 28, 32) = ?

step 2 Arrange the given integers in the horizontal form with space or comma separated format:
21, 28 and 32

step 3 Choose the divisor which divides each or most of the given integers (21, 28 and 32), divide each integers separately and write down the quotient in the next line right under the respective integers. Bring down the integer to the next line if any integer in 21, 28 and 32 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:

 2 21 28 32 2 21 14 16 2 21 7 8 2 21 7 4 2 21 7 2 3 21 7 1 7 7 7 1 1 1 1

step 4 Multiply the divisors to find the lcm of 21, 28 and 32:
= 2 x 2 x 2 x 2 x 2 x 3 x 7
= 672
LCM(21, 28, 32) = 672

The least common multiple for three numbers 21, 28 and 32 is 672 