# Formulas

## Number Sets # Natural Numbers

Set of natural numbers: $$\mathbb{N}$$
Set of natural numbers with zero: $$\mathbb{N_0}$$
Natural numbers: $$n$$, $$k$$, $$a$$, $$b$$, $$c$$
Arabic numerals: $$0$$, $$1$$, $$2$$, $$3$$, $$4$$, $$5$$, $$6$$, $$7$$, $$8$$, $$9$$
Roman numerals: $$I$$, $$V$$, $$X$$, $$L$$, $$C$$, $$D$$, $$M$$
1. Natural numbers $$\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}$$
These numbers can be used for counting and ordering.
2. Natural numbers including zero $$\mathbb{N_0} = \left\{ {0,1,2,3, \ldots } \right\}$$
These numbers can be used for indicating the number of objects.
3. Commutativity of addition $$a + b = b + a$$
4. Associativity of addition $$a + \left( {b + c} \right) = \left( {a + b} \right) + c$$
5. $$a + 0 = a$$
6. Commutativity of multiplication $$a \cdot b = b \cdot a$$
7. Associativity of multiplication $$a \cdot \left( {b \cdot c} \right) =$$ $$\left( {a \cdot b} \right) \cdot c$$
8. Distributivity of multiplication over addition $$a \cdot \left( {b + c} \right) =$$ $$a \cdot b + a \cdot c$$
9. $$a \cdot 0 = 0$$
10. $$a \cdot 1 = a$$
11. Even numbers $$n = 2k,\;\left( {n, k \in \mathbb{N}} \right)$$
12. Odd numbers $$n = 2k + 1,\;\left( {n, k \in \mathbb{N}} \right)$$
13. Roman numerals
14. Roman numerals $$1$$ to $$100$$
15. Divisibility by $$2$$ A number is divisible by $$2$$ if its last digit is divisible by $$2,$$ i.e. if the number is even.
16. Divisibility by $$3$$ A number is divisible by $$3$$ if the sum of its digits is divisible by $$3.$$
17. Divisibility by $$4$$ A number is divisible by $$4$$ if its last two digits are zero or form a number divisible by $$4.$$
18. Divisibility by $$5$$ A number is divisible by $$5$$ if its last digit is divisible by $$5,$$ i.e. equal to $$0$$ or $$5.$$
19. Divisibility by $$6$$ A number is divisible by $$6$$ if it is divisible by $$2$$ and by $$3,$$ i.e. if the number is even and the sum of its digits is divisible by $$3.$$
20. Divisibility by $$7$$ Subtract twice the last digit from the number formed by the remaining digits. If the result is divisible by $$7$$ then the original number is also divisible by $$7.$$ This method can be applied recursively.
21. Divisibility by $$8$$ A number is divisible by $$8$$ if its last three digits are zero or form a number divisible by $$8.$$
22. Divisibility by $$9$$ A number is divisible by $$9$$ if the sum of its digits is divisible by $$9.$$
23. Divisibility by $$10$$ A number is divisible by $$10$$ if its last digit is zero.