How to spot a perfect square in 30 seconds?

Written by Isha Chakraborty

Bio

2 mins read

First let us know what a perfect square is

A perfect square is a number that can be expressed as the product of two equal integers.

Example: 25 is a perfect square because it can be expressed as 5 * 5 (the product of two equal integers)
9 is a perfect square because it can be expressed as 3 * 3 (the product of two equal integers)
16 is a perfect square because it can be expressed as 4 * 4 (the product of two equal integers)

Now let’s get to the solution of the question we posted and  find out the perfect square number:

  1. The first number, 9492 cannot be a perfect square since the units digit is 2. No perfect square ends with 2 at the units place (you can mentally check that no number between 0 to 9 has a square that ends with 2).
  2. The second number, 4488 cannot be a perfect square since the units digit is 8. No perfect square ends with 8 at the units place (you can mentally check that no number between 0 to 9 has a square that ends with 8).
  3. The third number, 1211 cannot be a perfect square since the square of a natural number other than one is either a multiple of 3 or exceeds a multiple of 3 by 1. If you divide 1211 by 3 then you get 2 as the remainder.
  4. This leaves 1521, which is a square of 39. To check this fact quickly, follow the steps given below:

Step 1: Since 1521 ends with 1 as the units digit, the units digit of the square root will be either 1 or 9 (check for yourself that the only the square of a number ending with 1 or 9 will have 1 at the units place). Therefore, the square root will be of the form:

 _ 1 or _9

Step 2: Leave the last 2 digits of 1521 and focus on 15. The perfect square that is immediately preceding 15 is 9. 9 is the square of 3. Therefore, the square root will be:

31 or 39. 

Step 3: Now, take a number that lies exactly between 31 and 39. This number is 35. The square of 35 can be found out quickly:

35*35 = (30+5)(30+5) = 30*30 + 5*5 + 2*30*5 = 900 + 25 + 300 = 1225

Now, 1521>1225 and therefore, the square root of 1521 must be greater than the square root of 1225. Clearly, 39>35 but 31<35. Therefore, the square root of 1521 is 39.

Questions of this type frequently appear in different aptitude tests and competitive examinations. At Cuemath, we introduce the properties of perfect squares to our students early on to prepare them to ace their school exams and beyond.

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