Question

Find the least number of four digits which is a perfect number. Also, find the square root of the number so obtained.

Answer 1

Hint: For this question, we have to use the rules of finding a perfect square and we need to check if the remaining numbers have any square root or not. We all know that the smallest four digit number is $1000$. That is, the least four digit number is $1000$.
But, $1000$ is not a perfect square. Therefore, we shall check the successors to $1000$ which is $1001$, $1002$, $1003$ and so on.

Complete step-by-step solution:
Now, our question is to find the least four digit number which is a perfect square.
We shall check the last digit of the number that must be a perfect square (i.e.)$1,4,5,6,9 or 0$.
None of $1001,1004,1005,1006,1009$ are perfect squares and hence there are no perfect square numbers between $1000$ to $1009$.
And then, the ten-digit perfect square can’t be odd (i.e.)$1,3,5,7,9$. Therefore, the numbers between $1010$ to $1019$ are not perfect squares.
Also, $1020$ to $1023$ do not satisfy the above conditions.
Let us consider $1024$ and we check if it is a perfect square or not.
And, ${\left( {32} \right)^2} = 1024$
Then, $1024$ is a perfect square.
Therefore, the least four digit number which is a perfect square is $1024$.

Note: It is to be noted that squares of all integers are known as perfect square numbers.
We all know that the smallest four digit number is $1000$. That is, the least four digit number is $1000$.
There is another method to find the least four digit number which is a perfect square.
We know that $1000$ is not a perfect square.
Hence, the perfect squares of the smallest four digit number near to $1000$ must be checked.
${\left( {31} \right)^2} = 961$
${\left( {32} \right)^2} = 1024$
Therefore, the least four digit number which is a perfect square is $1024$.