What is the least number which is divisible by each of the numbers 12 18 and 24?

LCM of 12, 18 and 24 is equal to 72. The comprehensive work provides more insight of how to find what is the lcm of 12, 18 and 24 by using prime factors and special division methods, and the example use case of mathematics and real world problems.

what is the lcm of 12, 18 and 24?
lcm (12   18   24) = (?)
12 => 2 x 2 x 3
18 => 2 x 3 x 3
24 => 2 x 2 x 2 x 3

= 2 x 2 x 3 x 3 x 2
= 72
lcm (12, 18 and 24) = 72
72 is the lcm of 12, 18 and 24.

where,
12 is a positive integer,
18 is a positive integer,
72 is the lcm of 12, 18 and 24,
{2, 2, 3} in {2 x 2 x 3, 2 x 3 x 3, 2 x 2 x 2 x 3} are the most repeated factors of 12, 18 and 24,
{3, 2} in {2 x 2 x 3, 2 x 3 x 3, 2 x 2 x 2 x 3} are the the other remaining factors of 12, 18 and 24.

Use in Mathematics: LCM of 12, 18 and 24
The below are some of the mathematical applications where lcm of 12, 18 and 24 can be used:

1. to find the least number which is exactly divisible by 12, 18 and 24.
2. to find the common denominators for the fractions having 12, 18 and 24 as denominators in the unlike fractions addition or subtraction.

Use in Real-world Problems: 12, 18 and 24 lcm
In the context of lcm real world problems, the lcm of 12, 18 and 24 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 18 seconds and C tolls at 24 seconds repeatedly. The answer is that all bells A, B and C toll together at 72 seconds for the first time, at 144 seconds for the second time, at 216 seconds for the third time and so on.

Important Notes: 12, 18 and 24 lcm
The below are the important notes to be remembered while solving the lcm of 12, 18 and 24:

1. The repeated and non-repeated prime factors of 12, 18 and 24 should be multiplied to find the least common multiple of 12, 18 and 24, when solving lcm by using prime factors method.
2. The results of lcm of 12, 18 and 24 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.

For values other than 12, 18 and 24, use this below tool:

Hint: To find the LCM of the given number we need to find the multiples of each individual number and then we need to select the common multiple among all the three and then whichever number is the least is the lowest common multiple.

Complete step-by-step answer:
LCM stands for lowest common multiple. It is also known as lowest common divisor.
LCM can be found out by using the greatest divisor method, using prime factorization or by continuous division method.
Given, numbers are 12, 18, 24.
First write the multiple of all the three numbers,
$\Rightarrow$ Multiples of 12: 12 24 36 48 60 72
$\Rightarrow$ Multiples of 18: 18 36 54 72 90 108
$\Rightarrow$ Multiples of 24: 24 48 72 96 120 144
By the above observation it is clear that the 72 is the common multiple and it is also least common multiple. This method is known as the listing method.
The Lowest common multiple for 12, 18 and 24 is 72.

So, the correct answer is “Option A”.

Note: This question, can be solved by continuous division method as follows,
$
  2\left| \!{\underline {\,
  {12\,\,\,18\,\,\,24} \,}} \right. \\
  3\left| \!{\underline {\,
  {6\,\,\,\,\,\,\,9\,\,\,\,12} \,}} \right. \\
  2\left| \!{\underline {\,
  {2\,\,\,\,\,\,3\,\,\,\,\,\,4} \,}} \right. \\
  3\left| \!{\underline {\,
  {1\,\,\,\,\,\,\,\,3\,\,\,\,\,\,2} \,}} \right. \\
  2\left| \!{\underline {\,
  {1\,\,\,\,\,\,\,\,1\,\,\,\,\,\,2} \,}} \right. \\
  \,\,\,\,1\,\,\,\,\,\,\,\,1\,\,\,\,\,\,1 \\
 $
Now, multiply the divisors to get the L.C.M.
$\Rightarrow$ So, L.C.M. of 12, 18 and 24 will be equal to $2 \times 3 \times 2 \times 3 \times 2 \times 1 = 72$.
There is one more way to solve this problem by factor tree method, in this method we have to find all the factors of the given number, then by making the pairs we can find the L.C.M.

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