Hint: In this question, we need to find the least common multiple of four numbers 20, 36, 63, 67 using the division method. To find the LCM by division method, we write the given numbers in a row separately by commas, then divide the numbers by a common prime number. We stop dividing after reaching the prime numbers. The product of common and uncommon prime factors is the LCM of given numbers.
Complete step by step answer:
Here, we are given the numbers as 20, 36, 63, 67. Let us find their least common multiple using the division method.
First we write all numbers in a row separating them by a comma. Then we divide by a least prime number 2 which divides two of the numbers. Then we put the quotient directly under them and the number which does not get divided remains the same. After that, again dividing by 2, two of the numbers get divided and the other remains the same. Then dividing by 3, two numbers get divided. Again dividing by 3, two numbers get divided. Now dividing by 5, only one number gets divided. Similarly, dividing by 7 and 67. We get prime numbers by division method as 2, 2, 3, 3, 5, 7, 67.
\[\begin{align}
& 02\left| \!{\underline {\,
20,36,63,67 \,}} \right. \\
& 02\left| \!{\underline {\,
10,18,63,67 \,}} \right. \\
& 03\left| \!{\underline {\,
5,9,63,67 \,}} \right. \\
& 03\left| \!{\underline {\,
5,3,21,67 \,}} \right. \\
& 05\left| \!{\underline {\,
5,1,7,67 \,}} \right. \\
& 07\left| \!{\underline {\,
1,1,7,67 \,}} \right. \\
& 67\left| \!{\underline {\,
1,1,1,67 \,}} \right. \\
& 00\left| \!{\underline {\,
1,1,1,1 \,}} \right. \\
\end{align}\]
Now let us multiply these prime numbers we will get our required least common multiple.
Multiplying 2, 2, 3, 3, 5, 7, 67 we get, $ 2\times 2\times 3\times 3\times 5\times 7\times 67=84420 $ .
Therefore, the least common multiple of 20, 36, 63, 67 is 84420.
Note:
In the division method, students should make sure that we divide numbers by prime numbers only. If any number does not get divided at any stage then it will remain the same in the next row. We have to keep dividing until we get 1 for all numbers in the row.
