Question

How many non-square numbers lie between the squares of 12 and 13?

Answer 1

Hint: Here, we can see that 12 and 13 are consecutive numbers. So, all numbers between squares of 12 and 13 are non-square numbers. Therefore, first find squares of 12 and 13 and then subtract square of 12 from square of 13, we get numbers of non-square numbers. At the last subtract 1 from the result obtained as both extremes numbers are not included.

Complete step-by-step answer:
In these types of questions, a simple concept of numbers should be known that is between squares of two consecutive numbers all numbers are non-square numbers. Also one tricky point should remember that whenever we find the difference between two numbers we get a number of numbers between them including anyone of the extreme numbers. So we subtract 1 to exclude both extreme numbers.
Square of 12 = $ {12^2} = 144 $ and square of 12 = $ {13^2} = 169 $
As 12 and 13 are consecutive numbers so all numbers between their squares will be non-square numbers.
Therefore, 169 – 144 = 25
Total number of numbers between 169 and 144 (i.e., excluding 144 and 169) = 25 – 1 = 24.

Note: Alternate method: In simple case we can simply count the numbers present between the squares of two numbers, in this question, all non-square numbers between 144 and 169 are 145, 146, 147, 148 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168 169.
But a simple counting method will not be useful for large numbers as writing and counting all numbers between the two particular numbers. So, use this for small numbers.