LCM of 69 and 138
138 is the LCM of 69 and 138. 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 69 and 138, provided along with the use cases of mathematics and real-world problems.
what is the lcm of 69 and 138?
lcm (69 138) = (?)
69 => 3 x 23
138 => 2 x 3 x 23
= 3 x 23 x 2
= 138
lcm (69 and 138) = 138
138 is the LCM of 69 and 138.
where,
69 is a positive integer,
138 is a positive integer,
138 is the LCM of 69 and 138,
{3 x 23} in {3 x 23, 2 x 3 x 23} are the common factors of 69 and 138,
{2} in {3 x 23, 2 x 3 x 23} is the uncommon factors of 69 and 138.
Use in Mathematics: LCM of 69 and 138
The below are some of the mathematical applications where lcm of 69 and 138 can be used:
- to find the least number which is exactly divisible by 69 and 138.
- to find the common denominator for two fractions having 69 and 138 as denominators in the unlike fractions addition or subtraction.
In the context of lcm real world problems, the lcm of 69 and 138 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 69 seconds and bell B tolls at 138 seconds repeatedly. The answer is that all bells A and B toll together at 138 seconds for the first time, at 276 seconds for the second time, at 414 seconds for the third time and so on.
Important Notes: 69 and 138 lcm
The below are the important notes to be remembered while solving the lcm of 69 and 138:
- The common prime factors and the remaining prime factors of 69 and 138 should be multiplied to find the least common multiple of 69 and 138, when solving lcm by using prime factors method.
- The results of lcm of 69 and 138, and the lcm of 138 and 69 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 69 and 138?
The below solved example with step by step work shows how to find what is the lcm of 69 and 138 by using prime factors method and division method.
Solved example using prime factors method:
What is the LCM of 69 and 138?
step 1
Address the input parameters, values and observe what to be found:
Input parameters and values:
A = 69
B = 138
What to be found:
find the lcm of 69 and 138
step 2 Find the prime factors of 69 and 138:
Prime factors of 69 = 3 x 23
Prime factors of 138 = 2 x 3 x 23
step 3 Identify the repeated and non-repeated prime factors of 69 and 138:
{3, 23} are the most repeated factors and {2} is the non-repeated factor of 69 and 138.
step 4 Find the product of repeated and non-repeated prime factors of 69 and 138:
= 3 x 23 x 2
= 138
lcm(69 and 138) = 138
Hence,
lcm of 69 and 138 is 138
This special division method is one of the easiest ways to understand the entire calculation to find the LCM of 69 and 138.
step 1 Address the input parameters, values and observe what to be found:
Input parameters and values:
Integers: 69 and 138
What to be found:
lcm (69, 138) = ?
step 2 Arrange the given integers in the horizontal form with space or comma separated format:
69 and 138
step 3 Choose the divisor which divides each or most of the given integers (69 and 138), 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 69 and 138 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:
| 2 | 69 | 138 |
| 3 | 69 | 69 |
| 23 | 23 | 23 |
| 1 | 1 |
step 4 Multiply the divisors to find the lcm of 69 and 138:
= 2 x 3 x 23
= 138
LCM(69, 138) = 138
The LCM of two numbers 69 and 138 is 138