How to guess if the number is a perfect square ?


We have seen a number of shortcut to obtain the square of a number, to check if the number is a perfect square. In this tutorial, we will see how to guess if the number is a perfect square or not by just looking at the number itself.

In this tutorial, we will look at the shortcut, through which we can conclusively say that certain numbers are not perfect squares.

Rules :

  1. A perfect square will always have a unit digit of 1, 4, 5, 6, 9 or 0.
  2. A perfect square will always have a 10’s digit which is even, Except when the unit digit of a perfect square is 6, in which case the 10’s digit is always odd.
  3. If the unit digit of a perfect square is any other number, then the 10’s digit of a perfect square is always even.
  4. A perfect square ending in 5, will always have 2 as the 10’s digit.
  5. A perfect square ending in 0, will always have 0 as the 10’s digit ( even number of ending 0’s) .

Let us understand the shortcut with a simple example.

Example 1 : Consider the example 1438.

Solution : A perfect square can never have a unit digit of 2, 3, 8 or 7. Hence, we can conclude 1438 is not a perfect square.

Try Yourself


Can you identify which of the following numbers is not a perfect square ?

  1. 1029
  2. 3932
  3. 2008
  4. 9207


Example 2 :
Consider the example 3526.

Solution : Notice that the unit digit of the number in 3526. We have mentioned that, if the unit digit of a perfect square is 6, then the 10’s digit will always be odd. So, we can conclude 3526 is not a perfect square.

Important : It is very important to note that, this shortcut does not conclusively prove that the given number is a perfect square. However, by observing the number, you can conclusively say that the given number is not a perfect square based on above rules.


Example 3 :
Consider the example 4371.

Solution : Notice that the unit digit of the number is 4371. We have mentioned that, if the unit digit of a perfect square is 1, then the 10’s digit will always be even. However, in 4371, the 10’s digit is 7 which is an odd number. So, we can conclude 4371 is not a perfect square.

Can you identify which of the following numbers is not a perfect square ?

  1. 4891
  2. 9349
  3. 3999
  4. 3991 


Example 4 :
Consider the example 4345.

Solution : Notice that the unit digit of the number is 4345. We have mentioned that, if the unit digit of a perfect square is 5, then the 10’s digit will always be 2. However, in 4345, the 10’s digit is 4. So, we can conclude 4345 is not a perfect square.

So, 4325 could be perfect square, but 4335 is certainly not a perfect square.
 

Example 5 : Consider the example 56000.

Solution : Notice that the number 56000 has 3 0’s at the end. A perfect square with unit digit 0 will always have even number of 0’s at the end. Hence, we can conclude 56000 cannot be a perfect square.

Similarly 4900 could be a perfect square but 490 and 49000 is not a perfect square.

When does this shortcut not work ?


It is very important to note that, this shortcut does not conclusively prove that the given number is a perfect square. However, by observing the number, you can conclusively say that the given number is not a perfect square based on above rules.

For instance, notice that 1376 has a unit digit of 6 and 10’s digit 7 which is a odd number. However, 1376 is not a perfect square. As I mentioned earlier, this shortcut can be mainly used to conclusively say that the given number is not a perfect square based on above rules. However, we cannot use this shortcut to say if a number is a perfect square.

If you are not gone through my previous article on finding perfect squares based on Digital root method, I would advise, you to go through the article. You can read about the article here.

I hope, you find this article very useful. If you have any doubts or have any comments concerning this tutorial, Please leave a comment or contact me. I would be glad to help.

Kiran Chandrashekhar

Hey, Thanks for dropping by. My name is Kiran Chandrashekhar. I am a full-time software freelancer. I love Maths and Mathematical Shortcuts. Numbers fascinate me. I will be posting articles on Mathematical Shortcuts, Software Tips, Programming Tips in this website. I love teaching students preparing for various competitive examinations. Read my complete story.

4 Comments

  1. 3*3=9, 13*13=169. My point is that a perfect square can have a unit digit of 9, which is not stated!

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  3. If you are going for most excellent contents like myself,
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