Question

Find the least six-digit number which is a perfect square.

Answer 1

Hint: First, find the smallest six-digit number. Then find its square-root. When you obtain a square-root, check whether the square-root is an integer or not. If it is not an integer then find the nearest integer to the square-root that is greater than the square-root itself. You’ll have the required answer.

Complete step-by-step answer:
Here, we have to find the least six-digit number which is a perfect square.
So first we will find the least six-digit number or smallest six-digit number.
The smallest six digit number $ = 10000$
Now, we will find the square-root of this smallest six-digit number to determine whether the square-root is an integer or not.
$ \Rightarrow $ $\sqrt {100000} = \sqrt {10 \times 10 \times 10 \times 10 \times 10} $
The numbers which are in pairs are removed from under the square-root as they are perfect squares.
$ \Rightarrow $ $10 \times 10\sqrt {10} = 100\sqrt {10} $
Now, we will put the value of $\sqrt {10} $ which is $3.1623$
$ \Rightarrow $ $100 \times 3.1623 = 316.23$
The square-root we obtained is not an integer as it is in decimal.
So we have to find the nearest integer which is greater than the square-root itself.
So, the nearest integer $316.23$ is $317$ which is greater than the obtained square-root. Now, to find the perfect square we will multiply $317$by itself.
$ \Rightarrow $ $317 \times 317 = 100489$
$ \Rightarrow 317 \times 317 = 100489$
The least six-digit perfect square is $100489$.

Note: Here the student may go wrong if they find the integer smaller than the square-root. If we find the integer smaller than the square-root, then we will obtain a 5 digit perfect square by squaring the integer and we have to find a six-digit perfect square
 A perfect square is obtained when an integer is multiplied by itself. An integer is a number that is not a decimal or fraction. An integer can be both positive, negative, or zero.