Question

Find the greatest five-digit number which is a perfect square. Also, find the square root of the number so obtained.

Answer 1

Hint: In this problem first, we must find out which number is the greatest five-digit number. Then we evaluate whether this number is a perfect square or not. If it is a perfect square then we obtained our solution. But if it is not a perfect square then we have to evaluate the remainder which is excessive. Then this remainder is subtracted to obtain the desired number. After this, we can easily calculate the square root of this number.

Complete step by step answer:
In mathematics, the number system is the branch that deals with various types of numbers possible to form and easy to operate with different operators such as addition, multiplication and so on.
So, here we are required to find our greatest five-digit natural number. The greatest five-digit natural number is 99999.
Now considering the greatest five-digit natural number, we will try to evaluate whether it is a perfect square or not.
From the division method of square root evaluation, we get remainder as 143.
So, this confirms that 99999 is not a perfect square.
Now, subtracting 143 from 99999 we get, 99999 – 143 = 99856.
Hence, 99856 is the largest five- digit perfect square number.
For finding the square root of 99856 we could express 99856 in terms of prime factors as ${{2}^{4}}\times {{79}^{2}}$ and then taking the root, we get the square root of 99856 as:
$\begin{align}
  & \sqrt{{{2}^{4}}\times {{79}^{2}}}={{2}^{2}}\times 79 \\
 & =316 \\
\end{align}$

Finally, the largest five-digit perfect square number is 99856 whose square root is 316.

Note: The key step in solving this problem is the knowledge of the number system and different ways of representation of number. Another way of finding root is by hit and trial. Since, the last digit of square is 6 so the possible last digit of square root is 4 and 6. Also, the square of 300 is 90000, so the square root is greater than 300. The square of 310 is 96100, so the root is also greater than 310 but will be less than 320. So, possible roots are 314 and 316. Now, by calculating the square of 316 we obtain 99856.