Root Difference Inequality

$\left(\sqrt{n} - \sqrt{23\times 24}\right)^2<1$

Getting the square root of both sides, $|\sqrt{n} - \sqrt{23\times 24}|<1$
$-1< \sqrt{n} - \sqrt{23\times 24}<1$
$\sqrt{23\times 24}-1<\sqrt{n}<\sqrt{23\times 24}+1$

Squaring, $23\times 24-2\sqrt{23\times 24}+1<n<23\times 24+2\sqrt{23\times 24}+1$ Counting the number of integers n which satisfy the inequality above is similar to knowing how many integers satisfy: $-2\sqrt{23\times 24}<n-23 \times 24 <2\sqrt{23\times 24}$

Since, $23 < \sqrt{23\times 24}<23.5$ , thus $-47<n- 23 \times 24 <47$ .

Since $n$ is an integer, so $-46 \le n \le 46$ This gives us $46 - (-46) + 1 = 93$ integers.

mod(root(n)- root (23 24))&lt;=1 1-root (23 24)&lt;root(n)&lt; 1+root (23*24) 22.49468025&lt;root (n)&lt; 24.49468025 506.0106395&lt; n &lt; 599.9893606 total no of integers=599.9893606-506.0106395=93 Now, I agree with n=93!!!!!!!!!!!!!!!!!

Since n is an integer, the value (sqrt(n)-sqrt(23*24))^2 must be positive. Now, this value is less than 1 implies that its minimum value is 0.

Consider the case that the value of the given number is zero.

Then, n=23*24=552

Now consider the following equation (although it is not possible).

(sqrt(n)-sqrt(23*24))^2= 1

Note that 23 24 lies between 23^2 and 24^2. So, sqrt(23 24) lies between 23 and 24.

A small calculation reveals that the approximate value of sqrt(23*24) is 23.5.

So, ((sqrt(n) - 23.5))^2= 1 Or, sqrt(n) - 23.5= 1 [the negative case is dealt with later] Or, sqrt(n)= 24.5 Or, n= 600.25

But the given number is less than 1. This implies than n must be less than 600.25. Or, the maximum value of n is 600. So n can lie between 552 and 600.

Now consider the negative case [i.e sqrt(n) - 23.5 = -1)

sqrt(n)- 23.5 = -1 Or, sqrt(n) = 22.5 Or, sqrt(n)= 506.25

This implies that n can lie between 506 and 552.

Combining these two facts, we get that n must lie between 506 and 600 [both inclusive]. So the number of such integers is 600-506+1= 95

The inequality is equivalent to $\left( \sqrt{n} - \sqrt{23\times 24} +1\right) \left(\sqrt{n} - \sqrt{23\times 24} -1\right) < 0$ , or $\sqrt{23\times 24} -1 < \sqrt{n} < \sqrt{ 23\times 24} + 1$ . All the terms are greater than 0 so squaring everything, we get $23\times 24 - 2 \sqrt{23\times 24} +1 < n < 23\times 24 + 2 \sqrt{23\times 24} +1$ $\Rightarrow -2 \sqrt{23\times 24} < n - 23\times 24 - 1 < 2 \sqrt{23\times 24}$ . Since $23^2 < 23\times 24 < 23.5^2$ , we have $-46 \leq n - 23\times 24 - 1\leq 46$ , so there are 93 possible integer solutions.

According to me , there are two such integers - 23^2 (=529) and 24^2 (=576) as I have calculated as follows-

The quantity ( 23 24)^1/2 is approximately 23.49468 i.e 23.5 . Now, using modulus { n^1/2 - (23 24)^1/2 } &lt; 1, considering a straight line with points (23 24)^1/2 -1 and (23 24)^1/2 +1 , n^1/2 must strictly lie between these two numbers , i.e. n^1/2 must strictly lie between 23.5 - 1 and 23.5 + 1 i.e. n^1/2 must strictly lie between 22.5 and 24.5 i.e. n can attain values between (22.5) ^2 and (24.5)^2

                  Since n is demanded to be an integer , it can be either (23)^2  that is 529 
                                                                                or (24)^2 that is 576.