### 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) $$(-1)^3 = (-1) \times (-1) \times (-1) = -1$$