Question

Find the perfect square numbers between
A. 10 and 20
B. 50 and 60
C. 80 and 90

Answer 1

Hint: We know that 100 is the square of 10 which means all the squares of the numbers from 1 to 10 lie between 1 to 100, use this property to solve this question without any trouble.

Complete step by step solution:
Let us list the squares of all the numbers from 1 to 10 beforehand
\[\begin{array}{l}
{1^2} = 1\\
{2^2} = 4\\
{3^2} = 9\\
{4^2} = 16\\
{5^2} = 25\\
{6^2} = 36\\
{7^2} = 49\\
{8^2} = 64\\
{9^2} = 81\\
{10^2} = 100
\end{array}\]
Now let us see how many of these numbers lie between 10 and 20. Clearly 16 is the only number that lies between 10 and 20 and which is the square of 4 and hence the count of perfect squares between 10 and 20 is 1.
Now let us see how many of these numbers lie between 50 and 60 clearly we can observe that no numbers are present in between 50 and 60 which satisfies our condition though the closest are 49 and 64 but they don't belong to the given range, we are looking for and hence the count of perfect squares between 50 and 60 is 0.
Now let us see how many of these numbers lie between 80 and 90. Clearly only 81 lies between the numbers 80 and 90, which is a perfect square of 9 and hence the count of perfect squares between 80 and 90 is 1.

Note:
We can also do this question by examining each number in the intervals and checking the square root of 30 numbers but we can clearly understand that it will be a long method and chances of making mistakes is also high.