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We are familiar with a square very well. In a square, all the sides are equal in length and each angle is of measure 90as shown in the figure below.

A B

D C

Do you know why it is named as a square?It is named as a square because its area is a perfect square.Now, let us find the area of a square of side 15 cm.Area of the square = side side = 15 15 = 225Here, 225 is a perfect square.

Thus, perfect squares can be defined as follows:

“The result of the product of any natural number with itself is a perfect square or a square number”.

We can write the square of 27 as:272 = 27 27 = 729

In this way, we can write,12 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81102 = 100

From the above list of numbers, we can say that the numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfectsquares as these are the squares of the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, that are natural numbers.

A perfect square can also be defined as follows:

“A number x is said to be a perfect square, if x can be expressed as y2 where y is a natural number”.

Using this rule, let us check whether 35 is a perfect square or not.We have, 52 = 25 and 62 = 36.Also, 25 < 35 < 36

Identification Of Perfect Squares

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Here, we cannot express 35 as the square of any natural number. This is because 5 and 6 are consecutive naturalnumbers, and the square of 5 is less than 35 and the square of 6 is more than 35.

Therefore, we can say that 35 is not a perfect square.

In this way, we can check whether the given number is a perfect square or not. Let us discuss some more examples basedon perfect squares to understand the concept better.

Example 1:Find the perfect square number between 30 and 85.

Solution:The squares of 5, 6, 7, 8, 9, and 10 are as follows.52 = 2562 = 3672 = 4982 = 6492 = 81102 = 100Thus, the perfect squares between 30 and 85 are 36, 49, 64, and 81.

Example 2:Find the perfect squares between 70 and 90.

Solution:We know that for three consecutive natural numbers 8, 9, and 10,82 = 64, 92 = 81, and 102 = 100Also, 64 and 100 does not lie between 70 and 90. But, 81 lies between 70 and 90.Therefore, the only perfect square between 70 and 90 is 81.

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08M0602We know how to find the square of a number. On multiplying the same number two times, we obtain the square of thenumber. While calculating the squares of numbers, we come across various properties of perfect squares.

In order to understand the properties, let us calculate the square of the numbers between 10 and 20 and observe theproperties of perfect squares through observation.102 = 100112 = 121122 = 144132 = 169142 = 196152 = 225162 = 256172 = 289182 = 324192 = 361202 = 400

Property 1:In the above list of numbers,

10, 12, 14, 16, 18, and 20 are even numbers. We may observe that their squares, i.e., 100, 144, 196, 256,324, and 400 are also even numbers.

11, 13, 15, 17, and 19 are odd numbers and we may observe that their squares, i.e., 121, 169, 225, 289, and 361are also odd numbers.

We can generalize this property of perfect squares as follows:

“The squares of even numbers are even and the squares of odd numbers are odd.”

Using this rule, we can find whether the square of a given number is odd or even without squaring that number.For example, 657 is an odd number. Therefore, its square is also an odd number. Similarly, the square of 9028 is aneven number (as 9028 is an even number).

Property 2:Let us now look at the unit digits of the squares of the numbers. The following points can be observed.

Perfect squares always end with the numbers 0, 1, 4, 5, 6, and 9.Square of any number does not have numbers 2, 3, 7, or 8 at the end or as the last digit.

We can generalize this statement as:

“The unit place of a perfect square can never be 2, 3, 7, or 8”.

Using this rule, we can say that 512 is not a perfect square as this number ends with 2.

Properties Of Perfect Squares

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Property 3:By observing the last digit of a number, we can find the last digit of the square of that number. The following points will bevery helpful in finding the last digit of the square of any given number.(1) If a number ends with 1 or 9, then its square ends with 1.(2) If a number ends with 2 or 8, then its square ends with 4.(3) If a number ends with 3 or 7, then its square ends with 9.(4) If a number ends with 4 or 6, then its square ends with 6.(5) If a number ends with 5, then its square ends with 5.(6) If a number ends with 0, then its square ends with 0.

Using this rule, we can calculate the unit’s place of the square of a number without actually calculating its square.

For example, 548 ends with 8. Therefore, its square ends with 4.

Property 4:Let us find the squares of some multiples of 10, 100, and 1000.402 = 1600502 = 25007002 = 4900009002 = 810000

Here, we can observe that:(1) A perfect square always ends with an even number of zeroes.(2) If a number ends with n number of zeroes, then its square ends with 2n zeroes.

Using this rule, we can say that 25000 is not a perfect square as it ends with an odd number of zeroes. Using this rule, wecan also find the number of zeroes in the square of 3500000 without squaring it. Let us see how.Here, 3500000 ends with 5 zeroes. Therefore, its square ends with 2 × 5 = 10 zeroes.

Let us discuss some more examples to understand the concept better.

Example 1:Are 20124598 and 900000 perfect squares? Why?

Solution:20124598 is not a perfect square because a perfect square never ends with 8.900000 is also not a perfect square because it has odd number of zeroes.

Example 2:The squares of which of the following numbers are even numbers?9012, 3375, 1024, 378, 87

Solution:We know that the squares of even numbers are even.Thus, the squares of 9012, 1024, and 378 are even numbers.

Example 3:What would be the unit digit of the squares of the following numbers?8754, 967, 35120

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Solution:The square of 8754 ends with 6.We know that if a number ends with 4, then its square ends with 6. Therefore, the unit’s digit of the square of 8754 is 6.The square of 967 ends with 9.We know that if a number ends with 7, then its square ends with 9. Therefore, the unit’s digit of the square of 967 is 9.

The square of 35120 ends with 0.We know that if a number ends with 0, then its square also ends with 0. Therefore, the unit’s digit of the square of35120 is 0.