Question

Find the largest number of 5 digits, which is perfect square,
A. 99999
B. 99764
C. 99976
D. 99856

Answer 1

Hint: We will start by finding the largest number of 5 digits. Then we will use the fact that the number which is a perfect square has all the factors in even multiples and then find the factors of each of the options given in the question to find the greatest number which is a perfect square.

Complete step-by-step answer:
Now, we have to find the largest number of 5 digits which is a perfect square. We know the largest number of 5 digits is 99999.
Now, the prime factorization of 99999 is,
$99999={{3}^{2}}\times 271\times 41........\left( 1 \right)$
Now, we will take the next number which is less than 99999 and find its factor. So, in option (C) we have 99976
$99976=2\times 2\times 2\times 12497...........\left( 2 \right)$
Now, we have the next number which is smaller than 99999 is option (D) as 99856. Now, the prime factorization of this number is,
$\begin{align}
  & 99856=2\times 2\times 2\times 2\times 79\times 79 \\
 & 99856={{2}^{4}}\times {{79}^{2}}...........\left( 3 \right) \\
\end{align}$
Now, we know that for a number to be a perfect square its factors must be in even numbers as we can see from (1), (2) all the factors are not in even numbers but in (3) the factor 2 is 4 times and 79 is 2 times. Hence, all the factors are even times and therefore, we have the largest 5 digit number which is a perfect square is 99856.
Hence, the correct option is (D).

Note: To solve these questions it is important to note that we have used the fact that the number which is a perfect square has all its factors even number of times. Hence, we have the number 99856 as a perfect square because it has all the factors, even numbers of times.