Question

Find the smallest square number which is divisible by each of the numbers 4, 9 and 10

Answer 1

Hint- First, find the factors of the number that are the number when multiplied together, and give the original number.
In this question, first, find the factors to find the LCM and find the smallest number, which will make the LCM a perfect square number.


Complete step by step solution:
Find the factors of the numbers
\[
   2\underline {\left| {4,9,10} \right.} \\
   3\underline {\left| {2,9,5} \right.} \\
   2,3,5 \\
 \]
Hence the factors of each number are:
\[
  4 = 2 \times 2 \\
  9 = 3 \times 3 \\
  10 = 2 \times 5 \\
 \]
So the LCM of the three numbers will be:
\[LCM\left( {4,9,10} \right) = 2 \times 2 \times 3 \times 3 \times 5 = 180\]
Hence the LCM of numbers \[4,9,10\] is\[180\], where 180 is not a perfect number.
To make a perfect square number, all the factors of the number should be in pairs; hence we can conclude when the factors are multiplied by 5, it will make a square number
\[180 = \underline {2 \times 2} \times \underline {3 \times 3} \times 5\]
\[180 \times 5 = \underline {2 \times 2} \times \underline {3 \times 3} \times \underline {5 \times 5} = 900\]
Hence the number 900 is the smallest square number divisible by each number 4, 9, and 10.



Note: LCM of given numbers is exactly divisible by each of the numbers. During the LCM calculation, students must know the tables of various numbers, and they have to perform the operations step by step. Lastly, they have to multiply all the numbers by which they are dividing the given set of numbers. The least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by LCM (a, b), is the smallest positive integer that is divisible by both a and b.