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CUBES AND CUBE ROOTS 111

Consider a few numbers having 1 as the one’s digit (or unit’s). Find the cube of each
of them. What can you say about the one’s digit of the cube of a number having 1 as the
one’s digit?
Similarly, explore the one’s digit of cubes of numbers ending in 2, 3, 4, ... , etc.

TRY THESE

Find the one’s digit of the cube of each of the following numbers.
(i) 3331 (ii) 8888 (iii) 149 (iv) 1005
(v) 1024 (vi) 77 (vii) 5022 (viii) 53

7.2.1 Some interesting patterns
1. Adding consecutive odd numbers
Observe the following pattern of sums of odd numbers.
1 = 1 = 1 3
3 + 5 = 8 = 2 3
7 + 9 + 11 = 27 = 3 3
13 + 15 + 17 + 19 = 64 = 4 3
21 + 23 + 25 + 27 + 29 = 125 = 5 3
Is it not interesting? How many consecutive odd numbers will be needed to obtain
the sum as 10 ?
3

TRY THESE

Express the following numbers as the sum of odd numbers using the above pattern?
(a) 6 3 (b) 8 3 (c) 7 3
Consider the following pattern.
2 – 1 = 1 + 2 × 1 × 3
3
3
3 – 2 = 1 + 3 × 2 × 3
3
3
4 – 3 = 1 + 4 × 3 × 3
3
3
Using the above pattern, find the value of the following.
3
3
(i) 7 – 6 3 (ii) 12 – 11 3 (iii) 20 – 19 3 (iv) 51 – 50 3
3
3
2. Cubes and their prime factors
Consider the following prime factorisation of the numbers and their cubes.
Prime factorisation Prime factorisation each prime factor
of a number of its cube appears three times
in its cubes
4 = 2 × 2 4 = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2 × 2 3
3
3
3
3
6 = 2 × 3 6 = 216 = 2 × 2 × 2 × 3 × 3 × 3 = 2 × 3 3
15 = 3 × 5 15 = 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3 × 5 3
3
3
12 = 2 × 2 × 3 12 = 1728 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
3
=2 × 2 × 3 3
3
3