Page 54 - Start Up Mathematics_8 (Non CCE)

Conversely, property V implies that on dividing a natural number by 3, if it leaves the remainder as 0 or 1,
it may or may not be a perfect square. Further, if on division by 3, it leaves a remainder other than 0 or
1, then it is surely not a perfect square. For example, 538, 135, 73, etc. leave the remainder 0 or 1 when
divided by 3, but they are not perfect squares. Also, 260, 59, 71, etc. leave the remainder 2, when divided
by 3. So, they are not perfect squares.

VI. A perfect square when divided by 4, leaves the remainder as 0 or 1.
2
2
For example, (2) = 4, when divided by 4 leaves the remainder 0; (3) = 9, when divided by 4 leaves
the remainder 1.
Conversely, property VI implies that on dividing a natural number by 4, if it leaves the remainder as
0 or 1, it may or may not be a perfect square. Further, if on division by 4, it does not leave the remainder
as 0 or 1, then it is surely not a perfect square. For example, 50, 138, 354, etc. leave the remainder 2
when divided by 4. So, they are not perfect squares. Also, 296, 93, 328, etc. leave the remainder 0 or 1
when divided by 4. But they are not perfect squares.
VII. The square of a proper fraction is always smaller than the fraction.

1
3
For example, ʈ 2 = 1 ; 1 < 1 ʈ 2 = 9 ; 9 < 3 Units digit Units digit of
Á˜
Á˜
the square of
of the
˯
˯
2
4
16 16
4
4
2
4
number the number
VIII. The units digit of the square of a natural number is the square 0 0
of the units digit of the given number. In other words, there is a 1 or 9 1
relationship between the units digit of a number and that of its 2 or 8 4
square.
3 or 7 9
IX. For every natural number n, 4 or 6 6
2
2
(n + 1) – n = {(n + 1) + n}{(n + 1) – n} = (n + 1) + n 5 5
In other words, the difference in squares of any two consecutive
natural numbers is equal to their sum.
2
2
2
2
For example, 12 – 11 = 144 – 121 = 23 = 12 + 11, (35) – (34) = 1,225 – 1,156 = 69 = 35 + 34
2
X. For every natural number n, the sum of first n-odd natural numbers = n . In other words, the square of
a natural number n is equal to the sum of the first n-odd natural numbers,
2
i.e., n = 1 + 3 + 5 + 7 + ... + (2n – 1)
2
2
For example, 2 = 4 = 1 + 3, 3 = 9 = 1 + 3 + 5 and so on.
2
2
XI. There are no natural numbers a and b such that a = 2b . This property holds good in all the following
conditions:
(i) When a and b both are odd natural numbers.
For example, a = 3, b = 5
2
2
2
2
2
a = (3) = 9, b = (5) = 25, 2b = 2 × 25 = 50
2
Since, 9 π 50 \ a π 2b 2
(ii) When a is an odd and b is an even natural number.
For example, a = 3, b = 2 Do you know?
2
2
2
2
2
a = (3) = 9, b = (2) = 4, 2b = 2 × 4 = 8 The largest and smallest perfect
squares that use all the digits from
2
Since, 9 π 8 \ a π 2b 2 0 to 9 are
2
(iii) When a is an even and b is an odd natural number. (99066) = 9814072356
2
For example, a = 2, b = 3 (32043) = 1026753849
2
2
2
2
2
a = (2) = 4, b = (3) = 9, 2b = 2 × 9 = 18
2
Since, 4 π 18 \ a π 2b 2
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