Squares and square roots are opposite operations. The square of a number is found by multiplying the number by itself. For example, the square of 5 is 5 × 5 = 25. The square root of a number is the value which, when multiplied by itself, gives back the original number. For example, the square root of 25 is 5, because 5 × 5 = 25. Every positive number has two square roots: one positive and one negative. In maths, if a number is x, then its square is written as x² and its square root as √x.

In this math article we will study Perfect Square and the method to calculate in detail.

What is a Perfect Square?

Perfect Square is that number whose roots are rational. In other words if the square root of a number can be expressed in the p/q form, then the square number is known as a perfect square. For example if we consider a number such as 64, the square roots or 64 will be \( \pm8 \). This is because on multiplying 8 or -8 to itself we get 64. Moreover both 8 and -8 are rational numbers. Thus 64 is said to be a perfect square.

A perfect square can be related to the area of a geometric square shape as the name suggests. As all the sides of a square are equal in length, so by multiplying the value of one side to itself, we get the area of the square. Similarly on multiplying a number to itself, we get a perfect square. For example the perfect square for 6 is 36. This is because 6×6=36.

In the above image we can see how the perfect square of 5 is related with the area of a square of side length 5 units.

How to find Perfect Square Numbers?

The process to find perfect square numbers is the same as finding the square root of any number. We do the prime factorisation and get the roots. If there is only one root that may be positive or negative for a given number then the given number is a perfect square.

For example if we consider a number such as 81 and want to check whether it is a perfect square or not, then we first do the prime factorisation such as shown below.

From the above image we can write \( \sqrt{81}=\sqrt{3\times3\times3\times3}=\sqrt{9\times9}=\pm9 \)

By multiplying \( \pm9 \) to itself we get 81 and thus we find that 81 is a perfect square.

Perfect Square Formula

The process to find a perfect square is just to multiply a given number to itself. The formula to calculate the perfect square is given below.

In the above image X is the considered number for which we find the perfect number as N.

For example if we have X=11, then \( N=\left(11\right)^2=121 \)

Perfect Squares List 

The list of perfect squares for the first 100 natural numbers is given below.

How to Identify Perfect Squares?

A perfect square is a number that can be written as the product of a whole number multiplied by itself. One way to check whether a number could be a perfect square is by looking at its last digit (units place).

If you observe perfect squares from 1 to 20, you will notice an important rule:

• Perfect squares always end with 0, 1, 4, 5, 6, or 9.
• Numbers ending with 2, 3, 7, or 8 can never be perfect squares.

Here are some useful observations:

• Numbers ending in 3 or 7 will have their squares ending in 9.
• Numbers ending in 5 will always have their squares ending in 5.
• Numbers ending in 4 or 6 will always have their squares ending in 6.
• Numbers ending in 2 or 8 will always have their squares ending in 4.
• Numbers ending in 1 or 9 will always have their squares ending in 1.

By checking the last digit and applying these rules, you can quickly guess whether a number might be a perfect square.

Perfect Square between 1 to 100

Here we will see the perfect square numbers within the first 100 natural numbers as tabulated below.

Perfect Squares Between 1 and 1000

Between 1 and 1000, there are 30 perfect squares. These are:
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, and 961.

Each of these numbers can be written as the square of a whole number. For example, 400 = 20 × 20 and 729 = 27 × 27.

Properties of Perfect Squares

Perfect squares have some special features that make them different from other numbers. Here are the important properties:

1. Odd number of divisors:
   A perfect square always has an odd number of divisors. For example:
   • The divisors of 9 are 1, 3, and 9 (three divisors).
   • The divisors of 4 are 1, 2, and 4 (three divisors).
     For other numbers that are not perfect squares, the divisors always come in pairs, so their total number of divisors is even.
2. Last digit of perfect squares:
   A perfect square always ends with one of these digits: 0, 1, 4, 5, 6, or 9.
   It will never end with the digits 2, 3, 7, or 8.

These properties make it easy to check whether a number can be a perfect square or not.

Perfect Squares Tips and Tricks

The important points related to perfect squares are listed below.

• A perfect square that ends with zero digit will always have an even number of zeros at its end.
• Perfect squares are always positive.
• The square roots of a perfect square will be positive or negative.

Learn Square Root of 169.

Perfect Square Solved Examples

Problem 1: Find the perfect squares between 30 and 40.

Solution:

We see that between 30 and 40 we get a number 36 whose only square root is 6 i.e., if we multiply 6 by itself then we get 36. Thus 36 is the only number that is a perfect square between 30 and 40.

Problem 2: Find the greatest 4 digit number which is a perfect square.

Solution:

We get 9801 as the greatest 4 digit number that is a perfect square because \(\) \left(99\right)^2=9801 \(\).

Problem 3: Is 625 a perfect square?

Solution:

Yes 625 is a perfect square as we get the square root as 25. Thus \(\) \left(25\right)^2=625 \(\)

Problem 4: Is 1 a perfect square?

Solution:

Yes 1 is perfect as we discussed in the above given table.

Problem 5: Find the greatest 5 digit number which is a perfect square.

Solution:

We get 99856 as the greatest 5 digit number that is a perfect square because \(\) \left(316\right)^2=99856 \(\).

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