Question

Find the smallest square no. which is divisible by each of the numbers 8, 15 and 20

Answer 1

Hint- LCM is elaborated as Least Common Multiple, which is the lowest common factor among the integers. To find the LCM of the given numbers (Lowest common number), which is divisible by all numbers for which we are finding the LCM, the method includes basic factorization of the numbers to find factors that are multiplied together to form a 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
\[
   2\underline {\left| {8,15,20} \right.} \\
   2\underline {\left| {4,15,10} \right.} \\
   5\underline {\left| {2,15,5} \right.} \\
   2,3,1 \\
 \]
Hence the factors of each number are:
\[
  8 = 2 \times 2 \times 2 \\
  15 = 5 \times 3 \\
  20 = 2 \times 2 \times 5 \\
 \]
So the LCM of the three numbers will be:
\[LCM\left( {8,15,20} \right) = 2 \times 2 \times 2 \times 3 \times 5 = 120\]
Hence the LCM of numbers \[4,9,10\] is\[120\], where 120 is not a perfect number.
To make a perfect square number, all the factors should be in pairs hence we can conclude when the factors are multiplied by 2, 3, and 5, it will make a square number
\[120 = \underline {2 \times 2} \times 2 \times 3 \times 5\]
\[120 \times 2 \times 3 \times 5 = \underline {2 \times 2} \times \underline {2 \times 2} \times \underline {3 \times 3} \times \underline {5 \times 5} = 3600\]
Hence the number 3600 is the smallest square number divisible by each number 8, 15, and 20.


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.