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Solution: (a) 7777 is not a perfect square, as its unit or ones digit is 7. 7 7777

Justification: We can also verify this by prime factorisation. 11 1111
Since 7777 = 7 × 11 × 101, so we cannot form pairs of its prime factors. 101 101
1
Therefore, 7777 is not a perfect square.
(b) 968 is not a perfect square, as its ones digit is 8.
2 968
Justification: We can also verify this by prime factorisation. 2 484
Since 968 = 2 × 2 × 2 × 11 × 11, so we cannot form pair of its prime factors 2 242
as one 2 is left. 11 121
11 11
Therefore, 968 is not a perfect square. 1

Property 2

Observe the following:

2
2
2
10 = 100 100 = 10000 1000 = 1000000
2
2
2
20 = 400 Two zeros at 200 = 40000 Four zeros at 2000 = 4000000 Six zeros at the
the end
end
the end
2
2
2
30 = 900 300 = 90000 3000 = 9000000
We observe that perfect square numbers have only even numbers of zeros at the end. That means
a number ending in an odd number of zeros is never a perfect square.
Thus, we can say that the number of zeros at the end of a perfect square number is always even.
Example 8: Which among the following is a perfect square? Justify your answer.
(a) 50 (b) 4000 (c) 40000
Solution: (a) In 50, the number of zeros is 1(odd). So, 50 is not a perfect square.

(b) In 4000, the number of zeros is 3(odd). So, 4000 is not a perfect square.
(c) In 40000, the number of zeros is 4(even), and the number obtained after leaving
2
zeros is 4 which is 2 ¥ 2 = 2 . So, 40000 is a perfect square.

Note: 200 is not a perfect square number, even it ends with an even number of zeros, as 200 = 2 × 10 × 10,
the prime factor 2 is left unpaired.
Property 3

Observe the following

2
2
2 = 4 1 = 1
2
2
4 = 16 3 = 9
The square of an The square of an odd
6 = 36 even number is also 5 = 25 number is also an
2
2
an even number. odd number.
2
2
8 = 64 11 = 121
2
10 = 100 15 = 225
2
Thus, we can say that,
Squares of even numbers are even, and squares of odd numbers are odd.
127 Squares and Square Roots