# Formulas and Tables

Number Sets# Natural Numbers

Set of natural numbers with zero: \(\mathbb{N_0}\)

Natural numbers: \(n\), \(k\), \(a\), \(b\), \(c\)

Roman numerals: \(I\), \(V\), \(X\), \(L\), \(C\), \(D\), \(M\)

- Natural numbers \(\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)

These numbers can be used for counting and ordering. - 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. - Commutativity of addition \(a + b = b + a\)
- Associativity of addition \(a + \left( {b + c} \right) = \left( {a + b} \right) + c\)
- \(a + 0 = a\)
- Commutativity of multiplication \(a \cdot b = b \cdot a\)
- Associativity of multiplication \(a \cdot \left( {b \cdot c} \right) =\) \( \left( {a \cdot b} \right) \cdot c\)
- Distributivity of multiplication over addition \(a \cdot \left( {b + c} \right) =\) \( a \cdot b + a \cdot c\)
- \(a \cdot 0 = 0\)
- \(a \cdot 1 = a\)
- Even numbers \(n = 2k,\;\left( {n, k \in \mathbb{N}} \right)\)
- Odd numbers \(n = 2k + 1,\;\left( {n, k \in \mathbb{N}} \right)\)
- Roman numerals

- Roman numerals \(1\) to \(100\)

- Divisibility by \(2\)

A number is divisible by \(2\) if its last digit is divisible by \(2,\) i.e. if the number is even. - Divisibility by \(3\)

A number is divisible by \(3\) if the sum of its digits is divisible by \(3.\) - Divisibility by \(4\) A number is divisible by \(4\) if its last two digits are zero or form a number divisible by \(4.\)
- Divisibility by \(5\)

A number is divisible by \(5\) if its last digit is divisible by \(5,\) i.e. equal to \(0\) or \(5.\) - 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 \(3.\) - Divisibility by \(7\)

Sum 3 times the tens digit and the ones digit. If the result is divisible by \(7\) then the number is also divisible by \(7.\) - Divisibility by \(8\)

A number is divisible by \(8\) if its last three digits are zero or form a number divisible by \(8.\) - Divisibility by \(9\)

A number is divisible by \(9\) if the sum of its digits is divisible by \(9.\) - Divisibility by \(10\)

A number is divisible by \(10\) if its last digit is zero.