LCM of 12, 16 and 19
912 is the LCM of 12, 16 and 19. 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 12, 16 and 19, provided along with the use cases of mathematics and real-world problems.
what is the lcm of 12, 16 and 19?
lcm (12 16 19) = (?)
12 => 2 x 2 x 3
16 => 2 x 2 x 2 x 2
19 => 19
= 2 x 2 x 3 x 2 x 2 x 19
= 912
lcm (12, 16 and 19) = 912
The LCM of 12, 16 and 19 is 912
where,
12 is a positive integer,
16 is a positive integer,
912 is the LCM of 12, 16 and 19,
{2, 2} in {2 x 2 x 3, 2 x 2 x 2 x 2, 19} are the most repeated factors of 12, 16 and 19,
{3, 2, 2, 19} in {2 x 2 x 3, 2 x 2 x 2 x 2, 19} are the the other remaining factors of 12, 16 and 19.
Use in Mathematics: LCM of 12, 16 and 19
The below are some of the mathematical applications where lcm of 12, 16 and 19 can be used:
- to find the least number which is exactly divisible by 12, 16 and 19.
- to find the common denominators for the fractions having 12, 16 and 19 as denominators in the unlike fractions addition or subtraction.
In the context of lcm real world problems, the lcm of 12, 16 and 19 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 12 seconds, B tolls at 16 seconds and C tolls at 19 seconds repeatedly. The answer is that all bells A, B and C toll together at 912 seconds for the first time, at 1824 seconds for the second time, at 2736 seconds for the third time and so on.
Important Notes: 12, 16 and 19 lcm
The below are the important notes to be remembered while solving the lcm of 12, 16 and 19:
- The repeated and non-repeated prime factors of 12, 16 and 19 should be multiplied to find the least common multiple of 12, 16 and 19, when solving lcm by using prime factors method.
- The results of lcm of 12, 16 and 19 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 12, 16 and 19?
The below solved example with step by step work shows how to find what is the lcm of 12, 16 and 19 by using either prime factors method and special division method.
Solved example using prime factors method:
What is LCM of 12, 16 and 19?
step 1
Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 12
B = 16
C = 19
What to be found:
Find lcm of 12, 16 and 19
step 2 Find the prime factors of 12, 16 and 19:
Prime factors of 12 = 2 x 2 x 3
Prime factors of 16 = 2 x 2 x 2 x 2
Prime factors of 19 = 19
step 3 Identify the repeated and non-repeated prime factors of 12, 16 and 19:
{2, 2} are the most repeated factors and {3, 2, 2, 19} are the non-repeated factors of 12, 16 and 19.
step 4 Find the product of repeated and non-repeated prime factors of 12, 16 and 19:
= 2 x 2 x 3 x 2 x 2 x 19
= 912
LCM(12, 16 and 19) = 912
Hence,
The Least Common Multiple of 12, 16 and 19 is 912
This special division method is one of the easiest ways to understand the entire calculation to find the LCM of 12, 16 and 19.
step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 12, 16 and 19
What to be found:
lcm (12, 16, 19) = ?
step 2 Arrange the given integers in the horizontal form with space or comma separated format:
12, 16 and 19
step 3 Choose the divisor which divides each or most of the given integers (12, 16 and 19), 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 12, 16 and 19 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:
| 2 | 12 | 16 | 19 |
| 2 | 6 | 8 | 19 |
| 2 | 3 | 4 | 19 |
| 2 | 3 | 2 | 19 |
| 3 | 3 | 1 | 19 |
| 19 | 1 | 1 | 19 |
| 1 | 1 | 1 |
step 4 Multiply the divisors to find the lcm of 12, 16 and 19:
= 2 x 2 x 2 x 2 x 3 x 19
= 912
LCM(12, 16, 19) = 912
The LCM of three numbers 12, 16 and 19 is 912