How to check if the number is a Perfect Square



In this tutorial, we are going to learn the a very interesting Mathematical shortcut how to check if the number is a perfect square. This article is a follow-up to the article on How to obtain the square root of a perfect square? As you might have noticed, the shortcut in the previous article, it works only if the given number is a perfect square.

Steps


Step 1 : A perfect square never does in digit 2 3 7 8

This is the first observation you will make to check if the number is a perfect square or not. For example, consider the example 15623.

unit_digit


By just noticing the number itself, we can conclude that 15623 cannot be a perfect square. We do not have to go to Step 2.


Step 2 : obtain the digital root of the number.

Please go through my article on Digital Roots here. On a nutshell, digital root is just the sum of all the digits of the number.

Now, how does the digital root of a number would help in determining if a number is a perfect square or not. It turns out, a perfect square will always have a digital root of 0, 1, 4 or 7.

Let me explain it with an example.

Considering the example 15626. This number ends in digits 6. So it satisfies Step 1. But still we cannot conclude, this number as a perfect square.

Lets take the digital root of this number.

digital-root-square-root


The digital root of this number is 2. A perfect square will never have a digital root 2. Hence, we can conclude 15626 is not a perfect square.

Very easy right 🙂


Now, there is a rider for this shortcut though, even if both Steps are satisfied, that does not guarantee that the number is a perfect square.

Let me take up an example here. Consider the number 623461, which is not a perfect square.

digital-root-square-root-1


Notice that the unit digit is 1. This number could be a perfect square. Let us take the digital root.

digital-root-square-root-2


The digital root of 623461 is 4. So it satisfies both Step 1 and 2. However, we could not however conclude that 623461 is a perfect square though.


However, this shortcut comes in really handy to eliminate obvious choices which are not a perfect square to solve competitive examination where you need to find the perfect squares.



Using this shortcut, can you obtain the numbers which are not perfect squares?

  • 15129
  • 72253
  • 1296
  • 5041
  • 28928



Please contact me if some point in the tutorial is not clear. 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.

12 Comments

    • Thanks Akshay for your nice words. I am glad you liked it.
      ~Kiran

  1. I think you missed one more digit for digital root concept
    A perfect square must have digital roots as : 0,1,4,7, or “9” too?

    • Hello Rajan,

      Thanks for pointing it out. I knew in my head but could not write it. I will correct it.

    • Hello Rajan,

      I am sorry, I could not get back to you. The digital root of any number which ends in 9, is equivalent to digital root of 0. Hence, We do casting of 9’s

  2. is 5896 a square no
    5+8+9+6 = 28 => 2+8 =10=> 1+0= 1

    casting 9 out 5+8+6 = 19=> 1+9 =10 = 1+0 =1

    but calculating 5896 is not a perfect square no

  3. hi thank you…. can u plz solve this?
    1726 is perfect sq or not?
    step1: ends with 6 so step1 satisfies
    step2: digital root is 7 satisfies.
    “but it is not perfect square….”

    a number should satisfy both or any one condition.. pl explain … thanks in advance

  4. Short cuts for finding square of larger numbers like 8 digits. My homework is square for 20802721

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