Question

Find the lowest common multiple of 24, 36 and 40.
A. 120
B. 240
C. 360
D. 480

Answer 1

Hint: Lowest common multiple of three numbers is defined as the smallest number which is divisible by all of them. Lowest common multiple is commonly abbreviated as LCM. We can use the prime factorization method for finding LCM.

As per question, we have to find the lowest common multiple of 24, 36 and 40.
We have to find the smallest number which is divisible by 24, 36 and 40.
Let us use the prime factorization method for finding LCM of 24, 36 and 40.
Step 1:
Calculate prime factors of 24, 36 and 40.
Prime factorization of 24:
$\begin{align}
  & 2\left| \!{\underline {\,
  24 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6 \,}} \right. \\
 & \ \ 3 \\
\end{align}$
$\begin{align}
  & 24=2\times 2\times 2\times 3 \\
 & ={{2}^{3}}\times 3 \\
\end{align}$
Prime factorization of 36:
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & \ \ 3 \\
\end{align}$
$\begin{align}
  & 36=2\times 2\times 3\times 3 \\
 & ={{2}^{2}}\times {{3}^{2}} \\
\end{align}$
Prime factorization of 40:
$\begin{align}
  & 2\left| \!{\underline {\,
  40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  10 \,}} \right. \\
 & \ \ \ 5 \\
\end{align}$
$\begin{align}
  & 40=2\times 2\times 2\times 5 \\
 & ={{2}^{3}}\times 5 \\
\end{align}$
Step 2:
Write the prime factorization of each of them together.
$\begin{align}
  & 24=2\times 2\times 2\times 3 \\
 & 36=2\times 2\times 3\times 3 \\
 & 40=2\times 2\times 2\times 5 \\
\end{align}$
If a number is common, take it once in LCM and take the other remaining factors as they are;
$\begin{align}
  & LCM=\left( 2\times 2\times 2\times 3 \right)\times 3\times 5 \\
 & =360 \\
\end{align}$
Hence, LCM of 24, 36 and 40 is 360.

Note: Another method for finding LCM:
In each step, divide the numbers by their common factor. If no common factor exists, multiply all the divisors of each step and the remaining numbers in the last step to get an answer.
$\begin{align}
  & 2\left| \!{\underline {\,
  24,36,40 \,}} \right. \\
 & 2\left| \!{\underline {\,
  12,18,20 \,}} \right. \\
 & 2\left| \!{\underline {\,
  6,9,10 \,}} \right. \\
 & 2\left| \!{\underline {\,
  3,9,5 \,}} \right. \\
 & \ \ 1,3,5 \\
\end{align}$
If a number is not divisible by the divisor used in that step, take it same as it is the further step.
\[\begin{align}
  & LCM=2\times 2\times 2\times 3\times 3\times 5 \\
 & =360 \\
\end{align}\]