Let’s look at the cube of numbers from 1 to 50
Number | Cube |
1 | 1 |
2 | 8 |
3 | 27 |
4 | 64 |
5 | 125 |
6 | 216 |
7 | 343 |
8 | 512 |
9 | 729 |
10 | 1000 |
11 | 1331 |
12 | 1728 |
13 | 2197 |
14 | 2744 |
15 | 3375 |
16 | 4096 |
17 | 4913 |
18 | 5832 |
19 | 6859 |
20 | 8000 |
21 | 9261 |
22 | 10648 |
23 | 12167 |
24 | 13824 |
25 | 15625 |
30 | 27000 |
35 | 42875 |
40 | 64000 |
45 | 91125 |
50 | 125000 |
Let’s see some pattern in it, and find properties of cube numbers
Number of zeroes at the end of a perfect cube is always multiple of 3
So, number of zeroes at the end can be 3, 6, 9, 12, 15,....
Example
- 1,000 is a perfect cube
- 8,000 is a perfect cube
- 27,000,000 is a perfect cube
- 64,000,000,000 is a perfect cube
- 20 is not a perfect cube
- 400 is not a perfect cube
- 80,000 is not a perfect cube
Cube of even numbers are always even,
Cube of odd numbers are always odd
Example :
Cube of 2 is 8,
Cube of 6 is 216
And
Cube of 7 is 343
Cube of 9 is 729
Unit digit of cube of any number will be the unit digit of the cube of its last digit
Check Explanation
The cube of a negative integer is always negative
Example
(-1) ^{ 3 } = (-1) × (-1) × (-1) = -1
(-2) ^{ 3 } = (-2) × (-2) × (-2) = -8
The sum of the cubes of first in natural numbers is equal to the square of their sum
1 ^{ 3 } + 2 ^{ 3 } + 3 ^{ 2 } +..........+ n ^{ 3 } = (1 + 2 + 3 +..........+ n) ^{ 2 }
Example
1 ^{ 3 } + 2 ^{ 2 } + 3 ^{ 3 } + 4 ^{ 3 } = (1 + 2 + 3 + 4) ^{ 2 }
1 + 8 + 27 + 64 = (10) ^{ 2 }
100 = 100
Cubes of the number ending in digit 1, 4, 5, 6 and 9 are the numbers ending in the same digit
Number | Cube |
1 | 1 |
2 | 8 |
3 | 27 |
4 | 64 |
5 | 125 |
6 | 216 |
7 | 343 |
8 | 512 |
9 | 729 |
10 | 1000 |