Question

What is the greatest six digit number that is a perfect square?
(a) 998004
(b) 998006
(c) 998049
(d) 998001

Answer 1

Hint: We start solving the recalling the fact that the smallest seven-digit number 1000000 is a square of the number 1000, which is the smallest four-digit number. Using this, we understand that the square of the number that is just less than 1000 will give us the required answer. We then find the number that is 1 less than 100 and takes the square of it to get the required answer.

Complete step by step answer:
According to the problem, we need to find the greatest six-digit number which is a perfect square.
We know that the smallest seven-digit number is 1000000. We also know that 1000000 can be written as $ {{\left( 1000 \right)}^{2}} $ . We can also see that the number 1000 is the smallest four-digit number.
This tells us that the square of the number that is just less than 1000 gives the greatest six digit number which is a perfect square, which means that the square of the number 999 is the greatest six-digit number which is a perfect square.
So, we get $ {{999}^{2}}=998001 $ as the greatest six-digit number which is a perfect square.
 $\therefore$ The correct option for the given problem is (d).

Note:
Here we have considered only natural numbers to solve this problem otherwise, the answer would have been different. Whenever we get this type of problem, we try to find the smallest square of the next higher digit number which makes our process simpler. We can also solve this problem by prime factoring the given numbers and checking the factors which were perfect squares. Similarly, we can expect problems to find the greatest four-digit number that is a perfect square.