Question

The total non-square numbers that lie between the square of 30 and 31 are
(a) 59
(b) 61
(c) 60
(d) 62

Answer 1

Hint: We solve this problem simply by finding the number of numbers that lie between the square of 30 and 31 because there is not a single perfect square number between the square of 30 and 31. We know that all the square numbers are squares of some integers. In order to get a square number that lies between square of 30 and 31 that root of that number must lie between 30 and 31 which never happens. So, the total number of numbers that lie between squares of 30 and 31 are non-square numbers. We use the formula of finding the number of numbers between two numbers $$a,b$$as
 $$\Rightarrow N=a-b-1$$

Complete step-by-step answer:
We are asked to find the number of non-square numbers between squares 30 and 31.
Let us assume that there is a square number $${}^{\prime }{n}^{\prime }$$that lies between the square of 30 and 31.
Let us assume that a integer $${}^{\prime }{k}^{\prime }$$is the root of $${}^{\prime }{n}^{\prime }$$
Here, we can write
 $$k^2=n$$
We know that $${}^{\prime }{n}^{\prime }$$lie between the square of 30 and 31, so $${}^{\prime }{k}^{\prime }$$lies between 30 and 31.
Here, $${}^{\prime }{k}^{\prime }$$is integer but we have no integers between 30 and 31.
Therefore, we can say that our assumption of a square number that lies between the square of 30 and 31 is wrong.
We can say that there is not a single square number between 30 and 31. All the numbers are non-square numbers.
Now, let us find the number of numbers that lie between the square of 30 and 31 as follows
 $$\begin{array}{rl}& \Rightarrow N={31}^2-{30}^2-1\\ & \Rightarrow N=961-900-1\\ & \Rightarrow N=60\end{array}$$
Therefore, there are 60 non – square numbers between the square of 30 and 31.

So, the correct answer is “Option C”.

Note: Students will make mistakes in finding the number of numbers between two numbers.
The number of numbers between two numbers $$a,b$$as
 $$\Rightarrow N=a-b-1$$
But students will take it wrongly as
 $$\Rightarrow N=a-b$$
This will not give the correct answer because in this formula we include the second number $${}^{\prime }{b}^{\prime }$$, but we need numbers only in the between. So, we need to subtract ‘1’ to get numbers in the between only. This is the only mistake one can make.