Question

The LCM of 24, 36 and 40 is?
A. 4
B. 90
C. 360
D. 720

Answer 1

Hint: LCM is also known as Least Common multiple. LCM can be found for two and more than two numbers. Here we can find the LCM of 24, 36 and 40 using prime factorization. First we have to list out the prime factors of each of the given numbers. If factors have more than one occurrence, then we have to multiply them with each other, suppose we have a factor 3 and 2 common in the 3 numbers then consider 3, 2 only one time. And after all the repeated factors we have some factors which occur only once, multiply these factors to the previous product.

Complete step by step solution:
We are given to find the LCM of 24, 36 and 40.
First, we are finding their prime factorizations.
24 can be written as two times 12.
$24 = 2 \times 12$
12 can be written as two times 6.
$12 = 2 \times 6 \to 24 = 2 \times 2 \times 6$
6 can be written as two times 3.
$6 = 2 \times 3 \to 24 = 2 \times 2 \times 2 \times 3$
36 can be written as two times 18.
$36 = 2 \times 18$
18 can be written as two times 9.
$18 = 2 \times 9 \to 36 = 2 \times 2 \times 9$
9 can be written as three times 3.
$9 = 3 \times 3 \to 36 = 2 \times 2 \times 3 \times 3$
40 can be written as two times 20.
$40 = 2 \times 20$
20 can be written as two times 10.
$20 = 2 \times 10 \to 40 = 2 \times 2 \times 10$
10 can be written as two times 5.
$10 = 2 \times 5 \to 40 = 2 \times 2 \times 2 \times 5$
$
\Rightarrow {24} = 2{ \times 2}{ \times 2}{ \times 3} \\
\Rightarrow {36} = 2{ \times 2}{ \times 3}{ \times 3} \\
\Rightarrow {40} = 2{ \times 2}{ \times 2}{ \times 5}
$
As we can see in the first and second columns 2 is repeated, third column 2 repeated and 3 is not, fourth column 3 repeated and 5 is not.
Considering all the repeated factors once, non-repeated once
Therefore, LCM is $2 \times 2 \times 2 \times 3 \times 3 \times 5 = 360$
So, the correct answer is “Option C”.

Note: LCM is the least common multiple and GCD is the greatest common divisor. LCM is a common multiple of the numbers and GCD is the common divisor of the numbers. GCD is always less than or equal to LCM. GCD can also be calculated using prime factorization. Do not confuse LCM with GCD.