Page 57 - Start Up Mathematics_8 (Non CCE)

VIII. The natural numbers containing all digits as 6 with units digit as 7 follow another interesting pattern.
2
7 = 49
2
67 = 4489
2
667 = 444889
2
6667 = 44448889 and so on.
Example 3: Which of the following natural numbers are not perfect squares? Give reasons.

(a) 2,037 (b) 1,024 (c) 5,298 (d) 33,222 (e) 13,456
Solution: (a) 2,037 (c) 5,298 (d) 33,222 are not perfect squares because numbers ending
with digits 2, 3, 7, 8 are never perfect squares.
Example 4: Write the units digit of the squares of: (a) 61 (b) 372 (c) 2,134
Solution: (a) Since 61 has 1 at the units place, its square will have 1 as the units digit.
(b) Since 372 has 2 at the units place, its square will have 4 as the units digit.
(c) Since 2,134 has 4 at the units place, its square will have 6 as the units digit.
Example 5: Write the Pythagorean triplet, whose members are formed by (a) 10 and (b) 16.
Solution: (a) m = 10
2m = 2 × 10 = 20
2
2
m – 1 = (10) – 1 = 100 – 1 = 99
2
2
m + 1 = (10) + 1 = 100 + 1 = 101
So, the Pythagorean triplet is 20, 99, 101.
(b) m = 16
2m = 2 × 16 = 32
2
2
m – 1 = (16) – 1 = 256 – 1 = 255
2
2
m + 1 = (16) + 1 = 256 + 1 = 257
So, the Pythagorean triplet is 32, 255, 257.
Example 6: Without adding, find the following sum:
(a) 1 + 3 + 5 + 7 (b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17

Solution: (a) 1 + 3 + 5 + 7 is the sum of the first 4-odd natural numbers.
2
So, 1 + 3 + 5 + 7 = (4) = 16.
(b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 is the sum of the first 9-odd natural numbers.
2
So, 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = (9) = 81.
Example 7: Write (a) 25 and (b) 144 as a sum of odd natural numbers.
Solution: (a) 25 = 5 2

So, 25 can be written as a sum of the first 5-odd natural numbers.
\ 25 = 1 + 3 + 5 + 7 + 9
(b) 144 = (12) 2
So, 144 can be written as a sum of the first 12-odd natural numbers.
\ 144 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23
2
Example 8: Write 15 as the sum of two consecutive natural numbers.
2
2
2
Solution: (2n + 1) = (2n + 2n) + (2n + 2n + 1) where 2n + 1 is an odd natural number.
15 = 2 × 7 + 1 {Compare 2 × 7 + 1 with 2n + 1}
49