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Example 9: Determine whether the square of the following numbers would be odd or even.

(a) 522 (b) 677 (c) 169
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Solution: (a) (522) will be even, as the number 522 is an even number.
(b) (677) will be odd, as the number 677 is an odd number.
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(c) (169) will be odd, as the number 169 is an odd number.
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Example 10: The square of which of the following numbers would be an odd number?
551, 1058, 883, 6105
Solution: As we know that the square of an even number is always even, while the square of an
odd number is always odd. Among the given numbers, odd numbers are 551 and 883. Therefore,
the squares of 551 and 883 would be odd numbers.

Property 4

By observing the table of squares for numbers from 1 to 100, we find that:
• if a number ends with 1 or 9, then its square ends with 1.
Quick Check
• if a number ends with 4 or 6, then its square ends with 6.
Which of the following pairs of
• if a number ends with 2 or 8, then its square ends with 4. numbers have the same digit at the

• if a number ends with 3 or 7, then its square ends with 9. unit place in their squares?
(a) 24 and 38 (b) 35 and 50
• if a number ends with 5, then its square ends with 5. (c) 16 and 24 (d) 19 and 23

• if a number ends with 0, then its square ends with 0.
Example 11: Write the ones digit of the following squares.

(a) 219 2 (b) 555 2 (c) 893 2
Solution: (a) Ones digit of 219 is 9, so ones digit of 219 would be 1.
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(b) Ones digit of 555 is 5, so ones digit of 555 would be 5.
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(c) Ones digit of 893 is 3, so ones digit of 893 would be 9.
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Property 5

Observe the following:
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11 – 10 = 121 – 100 = 21 (Also, 11 + 10 = 21)
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17 – 16 = 289 – 256 = 33 (Also, 17 + 16 = 33)
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25 – 24 = 625 – 576 = 49 (Also, 25 + 24 = 49)
Thus, we can say that,
For every natural number n, the difference of squares of two consecutive natural numbers is
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equal to their sum, i.e., (n + 1) – n = (n + 1) + n, where n is a natural number.
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Example 12: Without finding the squares, find the value of 400 – 399 .
Solution: We know that the difference of the squares of two consecutive natural numbers is equal
to their sum.

Therefore, 400 – 399 = 400 + 399 = 799.
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