Question

Find the largest number of three digits which is a perfect square.

Answer 1

Hint: At first, we note down the largest three digit number which is $999$ and then we take the square root of it. The square root comes out as $31.606$ . We now take the greatest integer which is less than or equal to the number $31.606$ which is $31$ . Its square gives the required answer.

Complete step by step solution:
To find the greatest perfect square number which also happens to be a three digit number, we should know the greatest three digit number first. The greatest three digit number is $999$ . If we add the above number by one, we will get a four digit number. This means that, we can say that the largest number of three digits which is a perfect square must be less than or equal to the number $999$ . Now, if we start finding the square of the numbers randomly, then it will take too much time which is not possible in exam time. Thus, we will first find the square root of $999$ with the help of a calculator. So,
$\sqrt{999}=31.606$
Since the value of the square root is a decimal, so $999$ is not a perfect square. Now, if we take the number $32$ , then its square will definitely be a perfect square and it is greater than $999$ . So, this square becomes a four digit number. This means that we have to take the greatest integer which is less than or equal to the number $31.606$ . This number is $31$ and its square is ${{31}^{2}}=961$ .
Thus, we can conclude that the largest number of three digits which is a perfect square is $961$

Note: We can also find out the square root using the long division method. While claiming some result, we should always cross check the results with a calculator. Also, we should be careful enough to take the greatest integer which is less than or equal to the number $31.606$ as $31$ and not $30$ .