LCM of 30 and 105 is equal to 210. The comprehensive work provides more insight of how to find what is the lcm of 30 and 105 by using prime factors and special division methods, and the example use case of mathematics and real world problems.

__what is the lcm of 30 and 105?__

lcm (30 105) = (?)

30 => **2 x 3 x 5**

105 => **3 x 5 x 7**

= 3 x 5 x 2 x 7

= 210

lcm (30 and 105) = 210

**210 is the lcm of 30 and 105.**

__where,__

30 is a positive integer,

105 is a positive integer,

210 is the lcm of 30 and 105,

{3 x 5} in {2 x 3 x 5, 3 x 5 x 7} are the common factors of 30 and 105,

{2 x 7} in {2 x 3 x 5, 3 x 5 x 7} are the uncommon factors of 30 and 105.

__Use in Mathematics: LCM of 30 and 105__

The below are some of the mathematical applications where lcm of 30 and 105 can be used:

- to find the least number which is exactly divisible by 30 and 105.
- to find the common denominator for two fractions having 30 and 105 as denominators in the unlike fractions addition or subtraction.

In the context of lcm real world problems, the lcm of 30 and 105 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 30 seconds and bell B tolls at 105 seconds repeatedly. The answer is that all bells A and B toll together at 210 seconds for the first time, at 420 seconds for the second time, at 630 seconds for the third time and so on.

The below are the important notes to be remembered while solving the lcm of 30 and 105:

- The common prime factors and the remaining prime factors of 30 and 105 should be multiplied to find the least common multiple of 30 and 105, when solving lcm by using prime factors method.
- The results of lcm of 30 and 105, and the lcm of 105 and 30 are identical, it means the order of given numbers in the lcm calculation doesn't affect the results.

The below solved example with step by step work shows how to find what is the lcm of 30 and 105 by using prime factors method and division method.

__Solved example using prime factors method:__

What is the LCM of 30 and 105?

step 1
Address the input parameters, values and observe what to be found:

__Input parameters and values:__

A = 30

B = 105

__What to be found:__

find the lcm of 30 and 105

step 2 Find the prime factors of 30 and 105:

Prime factors of 30 = 2 x 3 x 5

Prime factors of 105 = 3 x 5 x 7

step 3 Identify the repeated and non-repeated prime factors of 30 and 105:

{3, 5} are the most repeated factors and {2 x 7} are the non-repeated factors of 30 and 105.

step 4 Find the product of repeated and non-repeated prime factors of 30 and 105:

= 3 x 5 x 2 x 7

= 210

lcm(30 and 105) = 210

Hence,

lcm of 30 and 105 is 210

This special division method is the easiest way to understand the entire calculation of what is the lcm of 30 and 105.

step 1 Address the input parameters, values and observe what to be found:

Integers: 30 and 105

lcm (30, 105) = ?

step 2 Arrange the given integers in the horizontal form with space or comma separated format:

30 and 105

step 3 Choose the divisor which divides each or most of the given integers (30 and 105), 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 30 and 105 is not divisible by the selected divisor; repeat the same process until all the integers are brought to 1 as like below:

2 | 30 | 105 |

3 | 15 | 105 |

5 | 5 | 35 |

7 | 1 | 7 |

1 | 1 |

step 4 Multiply the divisors to find the lcm of 30 and 105:

= 2 x 3 x 5 x 7

= 210

LCM(30, 105) = 210

The least common multiple for two numbers 30 and 105 is 210