How to check if a number is a Perfect Square

Squares of all integers are known as perfect squares. In this lesson, we will discuss a very interesting Mathematical shortcut: How to check whether a number is a perfect square or not. There are some properties of perfect squares which can be used to test if a number is a perfect square or not. They can definitely say if it is not the square. (i.e. Converse is not necessarily true).

All perfect squares end in 1, 4, 5, 6, 9 or 00 (i.e. Even number of zeros). Therefore, a number that ends in 2, 3, 7 or 8 is not a perfect square.

For all the numbers ending in 1, 4, 5, 6, & 9 and for numbers ending in even zeros, then remove the zeros at the end of the number and apply following tests:
  • Digital roots are 1, 4, 7 or 9. No number can be a perfect square unless its digital root is 1, 4, 7, or 9. You might already be familiar with computing digital roots. (To find digital root of a number, add all its digits. If this sum is more than 9, add the digits of this sum. The single digit obtained at the end is the digital root of the number.)
  • If unit digit ends in 5, ten’s digit is always 2.
  • If it ends in 6, ten’s digit is always odd (1, 3, 5, 7, and 9) otherwise it is always even. That is if it ends in 1, 4, and 9 the ten’s digit is always even (2, 4, 6, 8, 0).
  • If a number is divisible by 4, its square leaves a remainder 0 when divided by 8.
  • Square of even number not divisible by 4 leaves remainder 4 while square of an odd number always leaves remainder 1 when divided by 8.
  • Total numbers of prime factors of a perfect square are always odd.
4539 ends in 9, digit sum is 3. Therefore, 4539 is not a perfect square
5776 ends in 6, digit sum is 7. Therefore, 5776 may be a perfect square

Step 1: A perfect square never ends in 2, 3, 7 or 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.

15623

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
:

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.

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

Let’s take the digital root of this number.

1 5 6 2 6 = 5 + 6 = 11 = 1 + 1 = 2

So, 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.

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

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

6 2 3 4 6 1

The digital root of 623461 is 4. So it satisfies both Step 1 and 2. Still we cannot 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.

Is 14798678562 a perfect square? Is 15763530163289 a perfect square?

Examine both the units digits and the digital roots of perfect squares to help determine whether or not a given number is a perfect square.

How-to-check-if-a-number-is-a-perfect-squareAs we know a perfect square can only end in a 0, 1, 4, 5, 6, or 9; this should allow us to determine whether the first of our numbers is a perfect square. However, it isn't sufficient to draw a conclusion about the second number.

Again as we know that if a perfect square ends in 9; it’s tens digit is always even. Alas, even if we do this, it won't rule out numbers ending in 89, because '...89' is a possible square. 

However, as we know no number can be a perfect square unless its digital root is 1, 4, 7, or 9; so, find the digital root of our second number. It’s 5. As 5 isn't in this list, then the number is definitely not a perfect square.

So, we can conclude, a number cannot be an exact or perfect square if
-        it ends in 2, 3,7 or 8
-        it terminates in an odd number of zeros
-        its last digit is 6 but its penultimate (tens) digit is even
-        its last digit is not 6 but its penultimate (tens) digit is odd
-        its last digit is 5 but its penultimate (tens) digit is other than 2
-        its last 2 digits are not divisible by 4 if it is even number


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29 comments

  1. I read in the Guardian recently (Chris Maslanka's puzzles) that no perfect square can leave a remainder of 3 or 5 on division by 7. No explanation or justification was given. Do you know of any?

    ReplyDelete
    Replies
    1. Look at all possible remainders upon division by 7 (so 0 through 6) and see what the remainder is when you divide 0^2, 1^2, ..., 6^2 by 7. The possible remainders are 0, 1, 4, 2, 2, 4, and 1. So no perfect square gives a remainder of 3, 5, or 6 upon division by 7.

      Delete
    2. 679/7=97
      Reminder is 0 but it is not perfect square number

      Delete
  2. total number of prime factors is always even for a perfect square...total number of factors is odd...
    ex:36=2^2 * 3^2
    total no. of factors=(2+1)*(2+1)=6
    total number of distinct prime factors=2(2,3)
    correct me if i am wrong.thank you

    ReplyDelete
    Replies
    1. just a small typo (2+1)*(2+1)=9 as u said it always has to be odd hence its true

      Delete
  3. According to these rules are 61 and 76 perfect squares

    ReplyDelete
  4. 514 is a perfect square??...


    It's digital root is 1

    ReplyDelete
    Replies
    1. Nope, it's last 2 digits are not divisible by 4

      Delete
  5. 81 = 9^2?
    digital root is 9 so it fails on step 2
    Is the digital root the iterative sum of the digits until there is only one digit or is it the remainder when the number is divided by 9?

    ReplyDelete
  6. S siva:
    514%7=3, so its not a perfect square.

    ReplyDelete
  7. what about the no like 10,1000. these numbers are not perfect squares. This method fails at step II.

    ReplyDelete
    Replies
    1. You might not need to go to step 2 here. Step 1 rules out the possibility of perfect square as it ends with odd number of zeros.

      Delete
  8. What about 2356
    It satisfies unit digit , digital root, unit digit 6 and tens place odd no, last two digit divisible by 4 and even no.
    So where does ur method stand ?

    ReplyDelete
  9. 100856 IS PERFECT SQUARE OR NOT ?

    1. ENDS WITH 6
    2. DIGITAL ROOT = 2
    3. UNIT DIGIT =6 , AND TENS DIGIT IS ODD,
    4. LAST TWO DIGIT 56, IS DIVISIBLE BY 4.

    ANYONE CAN EXPLAIN ?

    ReplyDelete
  10. It's clearly stated that there are "riders" over and above these steps that these methods can only be used as a means of finding which number "cannot" be a perfect square ! Also, that even if one condition fails, the number fails to be a perfect square .. the conclusion sums it up perfectly -
    " A number cannot be a perfect square if ....... "

    ReplyDelete
  11. 15763530163289 is perfect square or not?

    ReplyDelete
  12. 1146600 it ends on 00, digital root is 9 but still it is not perfect square

    ReplyDelete
    Replies
    1. after removing 00 you get 6 as unit digit but tens digit(6) is even
      so it is not a perfect square.

      Delete
  13. Is there a method for binary numbers?

    ReplyDelete
  14. Is there a method for binary numbers?

    ReplyDelete
  15. 9981 is perfect square or not....can any one explain this correctly

    ReplyDelete
  16. Using this method we can only determine if a number is not a perfect square. That to only to an extent. You still need factorization to come to a final conclusion.

    ReplyDelete
  17. what about 4356
    As per above explanation
    step 1 satisfied
    step 2 not satisfied. Digital root is 9.
    But still it is a perfect square of 66
    I dont think this concept will work always.

    ReplyDelete

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