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Property 4

The cube of the numbers ending with digit 2 ends with digit 8 at its ones place and its vice versa.
3
3
For example, (12) = 12 × 12 × 12 = 1728, (18) = 18 × 18 × 18 = 5832
Property 5
The cube of the numbers ending with digit 3 ends with digit 7 at its ones place and its vice versa.

3
3
For example, (13) = 13 × 13 × 13 = 2197, (17) = 17 × 17 × 17 = 4913
Example 6: Find the digit in the ones place of the cube of each of the following numbers.
(a) 32 (b) 15 (c) 1026

Solution: (a) Ones digit of 32 = 2 Alternative Method:
And, the cube of 2 = 2 = 2 × 2 × 2 = 8 Last digit of 32 is 2, so by the property
3
So, the digit at ones place of cube of 32 is 8. of cube numbers the digit at the ones
place of cube of 32 is 8.

(b) Ones digit of 15 = 5 Alternative Method:
3
And, the cube of 5 = 5 = 5 × 5 × 5 = 125 Last digit of 15 is 5, so by the property
So, the digit at ones place of cube of 15 is 5. of cube numbers the digit at the ones
place of cube of 55 is 5.
(c) Ones digit of 1026 = 6 Alternative Method:
And, the cube of 6 = 6 = 6 × 6 × 6 = 216 Last digit of 1026 is 6, so by the
3
So, the digit at ones place of cube of 1026 is 6. property of cube numbers the digit
at the ones place of cube of 1026 is 6.

Example 7: Find the number of zeros in the cube of each of the following numbers without actual
multiplication.
(a) 50 (b) 200 (c) 1000

Solution: (a) For 50, we have one zero, so the number of zeros in the cube of 50 will be thrice of
one, i.e., 000 (3 zeros).

(b) For 200, we have two zeros, so the number of
zeros in the cube of 200 will be thrice of two, Think and Answer
i.e., 000000 (6 zeroes). Find the digit in the ones place of
(c) For 1000, we have three zeros, so the number of the cube of each of the following
zeros in the cube of 1000 will be thrice of three, numbers:
i.e., 000000000 (9 zeros). 1. 15 2. 48
Some Interesting Patterns of Cube Numbers

Just like square numbers, there are some interesting patterns occurs in cube numbers too.
1. Adding consecutive odd numbers
Observe the following pattern of sum of odd numbers:
3
1 = 1 = 1 Think and Answer
3

3 + 5 = 8 = 2
7 + 9 + 11 = 27 = 3 3 How many consecutive odd
13 + 15 + 17 + 19 = 64 = 4 3 numbers will be needed to obtain
3
the sum as 10 ?
21 + 23 + 25 + 27 + 29 = 125 = 5 3

153 Cubes and Cube Roots