LCM of 8, 9 and 21
504 is the LCM of 8, 9 and 21. The step-by-step solution using prime factors and special division methods is one of the simple methods to easily understand how to find what is the LCM of 8, 9 and 21, provided along with the use cases of mathematics and real-world problems.
what is the lcm of 8, 9 and 21?
lcm (8 9 21) = (?)
8 => 2 x 2 x 2
9 => 3 x 3
21 => 3 x 7
= 3 x 2 x 2 x 2 x 3 x 7
= 504
lcm (8, 9 and 21) = 504
The LCM of 8, 9 and 21 is 504
where,
8 is a positive integer,
9 is a positive integer,
504 is the LCM of 8, 9 and 21,
{3} in {2 x 2 x 2, 3 x 3, 3 x 7} is the most repeated factors of 8, 9 and 21,
{2, 2, 2, 3, 7} in {2 x 2 x 2, 3 x 3, 3 x 7} are the the other remaining factors of 8, 9 and 21.
Use in Mathematics: LCM of 8, 9 and 21
The below are some of the mathematical applications where lcm of 8, 9 and 21 can be used:
- to find the least number which is exactly divisible by 8, 9 and 21.
- to find the common denominators for the fractions having 8, 9 and 21 as denominators in the unlike fractions addition or subtraction.
In the context of lcm real world problems, the lcm of 8, 9 and 21 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 8 seconds, B tolls at 9 seconds and C tolls at 21 seconds repeatedly. The answer is that all bells A, B and C toll together at 504 seconds for the first time, at 1008 seconds for the second time, at 1512 seconds for the third time and so on.
Important Notes: 8, 9 and 21 lcm
The below are the important notes to be remembered while solving the lcm of 8, 9 and 21:
- The repeated and non-repeated prime factors of 8, 9 and 21 should be multiplied to find the least common multiple of 8, 9 and 21, when solving lcm by using prime factors method.
- The results of lcm of 8, 9 and 21 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.
How-to: What is LCM of 8, 9 and 21?
The below solved example with step by step work shows how to find what is the lcm of 8, 9 and 21 by using either prime factors method and special division method.
Solved example using prime factors method:
What is LCM of 8, 9 and 21?
step 1
Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 8
B = 9
C = 21
What to be found:
Find lcm of 8, 9 and 21
step 2 Find the prime factors of 8, 9 and 21:
Prime factors of 8 = 2 x 2 x 2
Prime factors of 9 = 3 x 3
Prime factors of 21 = 3 x 7
step 3 Identify the repeated and non-repeated prime factors of 8, 9 and 21:
{3} is the most repeated factor and {2, 2, 2, 3, 7} are the non-repeated factors of 8, 9 and 21.
step 4 Find the product of repeated and non-repeated prime factors of 8, 9 and 21:
= 3 x 2 x 2 x 2 x 3 x 7
= 504
LCM(8, 9 and 21) = 504
Hence,
The Least Common Multiple of 8, 9 and 21 is 504
This special division method is one of the easiest ways to understand the entire calculation to find the LCM of 8, 9 and 21.
step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 8, 9 and 21
What to be found:
lcm (8, 9, 21) = ?
step 2 Arrange the given integers in the horizontal form with space or comma separated format:
8, 9 and 21
step 3 Choose the divisor which divides each or most of the given integers (8, 9 and 21), 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 8, 9 and 21 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:
| 2 | 8 | 9 | 21 |
| 2 | 4 | 9 | 21 |
| 2 | 2 | 9 | 21 |
| 3 | 1 | 9 | 21 |
| 3 | 1 | 3 | 7 |
| 7 | 1 | 1 | 7 |
| 1 | 1 | 1 |
step 4 Multiply the divisors to find the lcm of 8, 9 and 21:
= 2 x 2 x 2 x 3 x 3 x 7
= 504
LCM(8, 9, 21) = 504
The LCM of three numbers 8, 9 and 21 is 504