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Step 4: The number will be a perfect square if there is no unpaired factor exists.

Let us consider a square number, say 25. 5 25
Resolving into prime factors, we find that 5 5
1
25 = 5 × 5 = 5 2
You can see that 25 can be grouped into a pair of equal factors, therefore 25 is a perfect square.

But there are some other natural numbers, which are not a product of two equal 2 200
numbers, i.e., not a perfect square. Let us consider a number, say, 200. 2 100

2 50
5 25
Resolving into prime factors, we find that 5 5

2
200 = 2 × 2 × 2 × 5 × 5 = 2 × 2 × 5 2 1
Here, 200 cannot be expressed as the product of pairs of equal prime factors, as 2 2 800
cannot be paired. Therefore, 200 is not a perfect square. Thus, a perfect square is 2 400
always expressed as the product of pairs of equal prime factors. 2 200
Example 2: Is 800 a perfect square? 2 100
2 50
Solution: Resolving into prime factors, we find that
5 25
2
2
800 = 2 × 2 × 2 × 2 × 2 × 5 × 5 = 2 × 2 × 2 × 5 2 5 5
Grouping the factors into pairs of equal factors, we find that 2 is left unpaired. 1
So, 800 is not a perfect square.

Example 3: Is 4225 a perfect square? 5 4225
5 845
Solution: Resolving into prime factors, we find that
13 169
4225 = 5 × 5 × 13 × 13 = 5 × 13 2 13 13
2
Since, 4225 can be grouped into pairs of equal factors, therefore, 1
4225 is a perfect square.
Quick Check
Example 4: Show that 11025 is a perfect square. Find the number
whose square is 11025. Find all perfect squares or
square numbers between:
Solution: Resolving into prime factors, we find that 1. 20 and 50 2. 70 and 100
11025 = 3 × 3 × 5 × 5 × 7 × 7 = 3 × 5 × 7 2
2
2
3 11025
Thus, we find that the prime factors of 11025 can be grouped into pairs and no 3 3675
factor is left unpaired. Therefore, 11025 is a perfect square. 5 1225
2
2
2
2
And, 11025 = 3 × 5 × 7 = (3 × 5 × 7) = 105 2 5 245
7 49
So, the square of 105 is 11025. 7 7
Example 5: Find the smallest number by which 1575 should be divided or 1
multiplied to get a perfect square.
3 1575
Solution: Resolving into prime factors, we find that 3 525
2
2
1575 = 3 × 3 × 5 × 5 × 7 = 3 × 5 × 7 5 175
5 35
Grouping the factors into pairs of equal factors, we find that 7 is left unpaired.
7 7
So, to get a perfect square, 1575 should be divided or multiplied by 7. 1

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