LCM of 9 and 84
252 is the LCM of 9 and 84. 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 9 and 84, provided along with the use cases of mathematics and real-world problems.
what is the lcm of 9 and 84?
lcm (9 84) = (?)
9 => 3 x 3
84 => 2 x 2 x 3 x 7
= 3 x 3 x 2 x 2 x 7
= 252
lcm (9 and 84) = 252
252 is the LCM of 9 and 84.
where,
9 is a positive integer,
84 is a positive integer,
252 is the LCM of 9 and 84,
{3} in {3 x 3, 2 x 2 x 3 x 7} is the common factors of 9 and 84,
{3 x 2 x 2 x 7} in {3 x 3, 2 x 2 x 3 x 7} are the uncommon factors of 9 and 84.
Use in Mathematics: LCM of 9 and 84
The below are some of the mathematical applications where lcm of 9 and 84 can be used:
- to find the least number which is exactly divisible by 9 and 84.
- to find the common denominator for two fractions having 9 and 84 as denominators in the unlike fractions addition or subtraction.
In the context of lcm real world problems, the lcm of 9 and 84 helps to find the exact time when two similar and recurring events 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 the bells A and B all toll together, if bell A tolls at 9 seconds and bell B tolls at 84 seconds repeatedly. The answer is that all bells A and B toll together at 252 seconds for the first time, at 504 seconds for the second time, at 756 seconds for the third time and so on.
Important Notes: 9 and 84 lcm
The below are the important notes to be remembered while solving the lcm of 9 and 84:
- The common prime factors and the remaining prime factors of 9 and 84 should be multiplied to find the least common multiple of 9 and 84, when solving lcm by using prime factors method.
- The results of lcm of 9 and 84, and the lcm of 84 and 9 are identical, it means the order of given numbers in the lcm calculation doesn't affect the results.
How-to: What is the LCM of 9 and 84?
The below solved example with step by step work shows how to find what is the lcm of 9 and 84 by using prime factors method and division method.
Solved example using prime factors method:
What is the LCM of 9 and 84?
step 1
Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 9
B = 84
What to be found:
find the lcm of 9 and 84
step 2 Find the prime factors of 9 and 84:
Prime factors of 9 = 3 x 3
Prime factors of 84 = 2 x 2 x 3 x 7
step 3 Identify the repeated and non-repeated prime factors of 9 and 84:
{3} is the most repeated factor and {3 x 2 x 2 x 7} are the non-repeated factors of 9 and 84.
step 4 Find the product of repeated and non-repeated prime factors of 9 and 84:
= 3 x 3 x 2 x 2 x 7
= 252
lcm(9 and 84) = 252
Hence,
lcm of 9 and 84 is 252
This special division method is one of the easiest ways to understand the entire calculation to find the LCM of 9 and 84.
step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 9 and 84
What to be found:
lcm (9, 84) = ?
step 2 Arrange the given integers in the horizontal form with space or comma separated format:
9 and 84
step 3 Choose the divisor which divides each or most of the given integers (9 and 84), 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 9 and 84 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:
| 2 | 9 | 84 |
| 2 | 9 | 42 |
| 3 | 9 | 21 |
| 3 | 3 | 7 |
| 7 | 1 | 7 |
| 1 | 1 |
step 4 Multiply the divisors to find the lcm of 9 and 84:
= 2 x 2 x 3 x 3 x 7
= 252
LCM(9, 84) = 252
The LCM of two numbers 9 and 84 is 252