Question

Find the LCM of 24, 60, 150 by fundamental theorem of arithmetic.

Answer 1

Hint: First we will look at what the fundamental theorem of arithmetic is, and then we will find the LCM of the given numbers using this theorem by dividing it by common factors one by one.

Complete step-by-step answer:

Fundamental theorem of arithmetic states that every integer greater than 1 is prime number or presented as a product of prime factorization also known as unique prime factorization.
We will write the factors of 24, 60, and 150.
$\begin{align}
  & 24=2\times 2\times 2\times 3 \\
 & 60=2\times 2\times 5\times 3 \\
 & 150=2\times 3\times 5\times 5 \\
\end{align}$
Now we are going to consider and take the highest number of times the factors of 24, 60 and 150 are appearing.
In 24, we have three 2’s which is the highest among all of them.
Now each number has one 3, therefore we will take only one 3.
And 150 has two 5’s which is the highest among all of them.
So, we have three 2’s, one 3 and two 5’s.
Hence we get,
$2\times 2\times 2\times 3\times 5\times 5=600$
Hence, 600 is the LCM of 24, 60, 150.

Note: Students can make mistakes by thinking that if we multiply all the given numbers then it will be the multiple of 24, 60, 150. But we have been asked to find the least common multiple not just the common multiple, so we have taken the common number only once in the consideration. So this point must be kept in mind.
We can also find the LCM using the given method below.
$\begin{align}
  & 6\left| \!{\underline {\,
  24,60,150 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4,10,25 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2,5,25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,5,25 \,}} \right. \\
 & 5\left| \!{\underline {\,
  1,1,5 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1,1,1 \,}} \right. \\
\end{align}$
Hence, from this we have LCM = $6\times 2\times 2\times 5\times 5=600$