Question

# Find the LCM of 72, 84 and 126.

Hint: To solve this question we should know the concept of LCM or least common multiple. We will find the LCM of 72, 84 and 126 using the division method. Basically, we will do the prime factorization in the first step of all the numbers and continue it till co - prime numbers are left in the last row. In the last step, we will multiply the co - prime numbers left in the last row with the prime number with which we had divided.

It is given in the question that we have to find the LCM of the numbers 72, 84 and 126. We will use the division method to solve this question. In order to find the LCM by using the division method, we need to follow the following steps.
Step 1: We will write all the given numbers in a horizontal line, separated by commas.
Step 2: We will then divide the numbers with a suitable prime number, which exactly divides at least two of the given numbers.
Step 3: We will put the quotient directly under the numbers in the next row. If the number above is not divided exactly, then we will write that number as it is in the next row.
Step 4: We will continue this process till co - prime are left in the last row.
Step 5: We will multiply all the prime numbers by which we had divided and the co - prime number left in the last row. The product is the LCM of the given number.
Now, in this question, the given numbers are 72, 84 and 126. So, let us find their LCM according to the steps discussed earlier.
Step 1:
$\left| \!{\underline {\, 72,84,126 \,}} \right.$
Step 2:
\begin{align} & 2\left| \!{\underline {\, 72,84,126 \,}} \right. \\ & \text{ }\left| \!{\underline {\, 36,42,63 \,}} \right. \\ \end{align}
Step 3:
\begin{align} & 2\left| \!{\underline {\, 72,84,126 \,}} \right. \\ & 2\left| \!{\underline {\, 36,42,63 \,}} \right. \\ & 3\left| \!{\underline {\, 18,21,63 \,}} \right. \\ & 3\left| \!{\underline {\, 6,7,21 \,}} \right. \\ & 7\left| \!{\underline {\, 2,7,7 \,}} \right. \\ & 2\left| \!{\underline {\, 2,1,1 \,}} \right. \\ & \text{ }1,1,1 \\ \end{align}
Step 4:
We have got the co - prime number.
Step 5:
\begin{align} & 2\times 2\times 3\times 3\times 7\times 2\times 1\times 1\times 1 \\ & 2\times 2\times 3\times 3\times 7\times 2 \\ & 504 \\ \end{align}
Thus, the LCM of 72, 84 and 126 is 504.

Note: The students often make mistakes in the last step. They may multiply one number only once like, we have 2 repeating three times and 3 repeating 2 times. And the students may take one 3 and one 2 by considering the pairing of numbers. But while finding the LCM, we have to multiply all the factors and pairing of numbers is not required.