Question

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

Answer 1

Hint: LCM (lowest common multiple) of the given numbers is divisible by all the given numbers. So, first of all we have to find the lowest common multiple (LCM) of the given numbers, then write the multiple of LCM and the first multiple of the LCM which is a perfect square, is the required smallest perfect square number which is divisible by the number \[4\], $9$ and $10$.

Complete step-by-step solution:
Given that, we have to find the smallest perfect square number which is divisible by each of the numbers $4$ , $9$ and $10$.
First of all, find the lowest common multiple of the given numbers $4$ , $9$ and $10$.
$
  2\left| {4,9,10} \right. \\
  \,\,\,\,\left| {\overline {2,9,5} } \right. \\
 $
LCM of \[4\], $9$ and $10$ $ = 2 \times 2 \times 9 \times 5$
LCM of \[4\], $9$ and $10$$ = 180$.
So, the lowest common multiple (LCM) of \[4\], $9$ and $10$ is $180$.
Now, we have to write the multiple of $180$.
$
  180 \times 1 = 180 \\
  180 \times 2 = 360 \\
  180 \times 3 = 540 \\
  180 \times 4 = 720 \\
  180 \times 5 = 900
$.
By writing the multiple we got that the first number which is a perfect square, is the fifth multiple of $180$. So, the smallest perfect square number which is divisible by each of the numbers \[4\], $9$ and $10$ is \[900\].

Hence, the required perfect square number is $900$.

Note: Lowest common multiple can be alternatively calculated by the prime factorization method in which firstly we have to write the prime factor of \[4\], $9$ and $10$. And then multiply the factors by writing the factors single time which is repeated and the factor which is present once.
Similarly, if we have to find the number which is completely divisible by the given numbers then we have to only calculate their LCM which is the required number.