Theory:

1. The cube of a positive number is always positive.
Example:
\(4^3 = 4 \times 4 \times 4 = 64\)
 
2. The cube of a negative number is always negative.
Example:
\((-4)^3 = (-4) \times (-4) \times (-4) = -64\)
 
3. The cube of every even number is even.
Example:
\(2^3 = 8\), \(4^3 = 64\), \(6^3 = 216\), \(8^3 = 512\), ...
 
Here, \(8\), \(64\), \(216\) and \(512\) are all even numbers.
 
4. The cube of every odd number is odd.
Example:
\(1^3 = 1\), \(3^3 = 27\), \(5^3 = 125\), \(7^3 = 343\), ...
 
Here, \(1\), \(27\), \(2125\) and \(343\) are all odd numbers.
 
5. If a natural number ends at \(0\), \(1\), \(4\), \(5\), \(6\) or \(9\), its cube also ends with the same \(0\), \(1\), \(4\), \(5\), \(6\) or \(9\), respectively.
Example:
(i) \(10^3 = 100\underline{0}\)
 
(ii) \(1^3 = \underline{1}\)
 
(iii) \(4^3 = 6\underline{4}\)
 
(iv) \(5^3 = 12\underline{5}\)
 
(v) \(6^3 = 21\underline{6}\)
 
(vi) \(9^3 = 72\underline{9}\)
 
6. If a natural number ends at \(2\) or \(8\), its cube ends at \(8\) or \(2\), respectively.
Example:
(i) \(2^3 = \underline{8}\)
 
(ii) \(8^3 = 51\underline{2}\)
 
7. If a natural number ends at \(3\) or \(7\), its cube ends at \(7\) or \(3\), respectively.
Example:
(i) \(3^3 = 2\underline{7}\) 
 
(ii) \(7^3 = 34\underline{3}\)
 
8. A perfect cube does not end with two zeroes.
Example:
\(10^3 = 1000\), \(20^3 = 8000\), …
 
9. The sum of the cubes of first \(n\) natural numbers is equal to the square of their sum.
 
That is, \(1^3 + 2^3 + 3^3 + 4 ^3 + …. + n^3 = (1 + 2 + 3 + 4 + … + n)^2\)
Example:
\(1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36\)
 
\((1 + 2 +3)^2 = 6^2 = 36\)
 
So, \(1^3 + 2^3 + 3^3 = (1 + 2 +3)^2\)
 
10. Each prime factor of a number appears three times in its cube.
Example:
\(6^3 = 216\)
 
Prime factor of \(6\) \(=\) \(2 \times 3\)
 
Prime factor of \(216\) \(=\) \((2 \times 2 \times 2) \times (3 \times 3 \times 3)\)
 
11. There are only three numbers whose cube is equal to itself.
 
(i) \(0^3 = 0 \times 0 \times 0 = 0\)
 
(ii) \(1^3 = 1 \times 1 \times 1 = 1\)
 
(iii) \((-1)^3 = (-1) \times (-1) \times (-1) = -1\)