9, 12 and 16 LCM
LCM of 9, 12 and 16 is equal to 144. The comprehensive work provides more insight of how to find what is the lcm of 9, 12 and 16 by using prime factors and special division methods, and the example use case of mathematics and real world problems.
what is the lcm of 9, 12 and 16?
lcm (9 12 16) = (?)
9 => 3 x 3
12 => 2 x 2 x 3
16 => 2 x 2 x 2 x 2
= 2 x 2 x 3 x 3 x 2 x 2
= 144
lcm (9, 12 and 16) = 144
144 is the lcm of 9, 12 and 16.
where,
9 is a positive integer,
12 is a positive integer,
144 is the lcm of 9, 12 and 16,
{2, 2, 3} in {3 x 3, 2 x 2 x 3, 2 x 2 x 2 x 2} are the most repeated factors of 9, 12 and 16,
{3, 2, 2} in {3 x 3, 2 x 2 x 3, 2 x 2 x 2 x 2} are the the other remaining factors of 9, 12 and 16.
Use in Mathematics: LCM of 9, 12 and 16
The below are some of the mathematical applications where lcm of 9, 12 and 16 can be used:
- to find the least number which is exactly divisible by 9, 12 and 16.
- to find the common denominators for the fractions having 9, 12 and 16 as denominators in the unlike fractions addition or subtraction.
In the context of lcm real world problems, the lcm of 9, 12 and 16 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 9 seconds, B tolls at 12 seconds and C tolls at 16 seconds repeatedly. The answer is that all bells A, B and C toll together at 144 seconds for the first time, at 288 seconds for the second time, at 432 seconds for the third time and so on.
Important Notes: 9, 12 and 16 lcm
The below are the important notes to be remembered while solving the lcm of 9, 12 and 16:
- The repeated and non-repeated prime factors of 9, 12 and 16 should be multiplied to find the least common multiple of 9, 12 and 16, when solving lcm by using prime factors method.
- The results of lcm of 9, 12 and 16 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 the LCM of 9, 12 and 16?
The below solved example with step by step work shows how to find what is the lcm of 9, 12 and 16 by using either prime factors method and special division method.
Solved example using prime factors method:
What is the LCM of 9, 12 and 16?
step 1
Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 9
B = 12
C = 16
What to be found:
find the lcm of 9, 12 and 16
step 2 Find the prime factors of 9, 12 and 16:
Prime factors of 9 = 3 x 3
Prime factors of 12 = 2 x 2 x 3
Prime factors of 16 = 2 x 2 x 2 x 2
step 3 Identify the repeated and non-repeated prime factors of 9, 12 and 16:
{2, 2, 3} are the most repeated factors and {3, 2, 2} are the non-repeated factors of 9, 12 and 16.
step 4 Find the product of repeated and non-repeated prime factors of 9, 12 and 16:
= 2 x 2 x 3 x 3 x 2 x 2
= 144
lcm(20 and 30) = 144
Hence,
lcm of 9, 12 and 16 is 144
This special division method is the easiest way to understand the entire calculation of what is the lcm of 9, 12 and 16.
step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 9, 12 and 16
What to be found:
lcm (9, 12, 16) = ?
step 2 Arrange the given integers in the horizontal form with space or comma separated format:
9, 12 and 16
step 3 Choose the divisor which divides each or most of the given integers (9, 12 and 16), 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, 12 and 16 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:
| 2 | 9 | 12 | 16 |
| 2 | 9 | 6 | 8 |
| 2 | 9 | 3 | 4 |
| 2 | 9 | 3 | 2 |
| 3 | 9 | 3 | 1 |
| 3 | 3 | 1 | 1 |
| 1 | 1 | 1 |
step 4 Multiply the divisors to find the lcm of 9, 12 and 16:
= 2 x 2 x 2 x 2 x 3 x 3
= 144
LCM(9, 12, 16) = 144
The least common multiple for three numbers 9, 12 and 16 is 144