Odd and Even Numbers
Integers that are divisible by 2 are called even numbers, whereas integers that are not divisible by 2 are called odd numbers.
Given $n \in \mathbb{Z}$, even numbers are algebraically represented by the general expression $2n$, and odd numbers are represented by $2n-1$ (or $2n+1$).
If we denote the set of odd numbers as $O$ and the set of even numbers as $E$, they can be listed as follows:
$$O = \{ \dots, -5, -3, -1, 1, 3, 5, \dots, 2n – 1, 2n + 1, \dots \}$$
$$E =\{\dots,−6,−4,−2,0,2,4,6,\dots,2n,\dots\} $$
Let $o$ represent any odd number and $e$ represent any even number. The arithmetic properties governing operations with odd and even numbers are outlined below:
\[o \pm o = e\]
\[o \pm e= t\]
\[e \pm o = t\]
\[ç \pm e= e\]
\[o \cdot o = t\]
\[o \cdot e = e\]
\[e \cdot o = e\]
\[e \cdot e = e\]
Furthermore, for any integer exponent $n \in \mathbb{N}^{+}$, it holds true that $t^n=o$ and $ç^n=e$.
Warning:
In any multiplication process, if the resulting product is an odd number, all individual factors must strictly be odd numbers. Conversely, if the product is an even number, at least one of the factors must be an even number.
Examples:
- If $3\cdot7^3\cdot5^{10}\cdot{11}=a$, then $a$ is an odd number.
- If $5^4\cdot7^9\cdot2^{11} = b$, then $b$ is an even number because the factor $2^{11}$ within the product is even.
Example:
Assuming $n$ is an integer, let us evaluate whether the parity (odd or even nature) of the following expressions can be definitively determined: $$6\cdot n^3 + 7^3, \quad 5^4-6^7+10\cdot n,\quad 2\cdot n^5+12,\quad 8\cdot n^6, \quad 11\cdot n^4, \quad n\cdot 7^3$$ Since any integer multiplied by an even coefficient yields an even product, the terms $$6\cdot n^3 ,\quad 10\cdot n,\quad 2\cdot n^5,\quad 8\cdot n^6 \quad \text{as well as} \quad 6^7$$ are always even. Furthermore, since base values $$7^3 \quad \text{and} \quad 5^4$$ are odd, we can evaluate the expressions as follows:
- $6\cdot n^3 + 7^3$ is an odd number $(\text{even} + \text{odd} = \text{odd})$
- $5^4-6^7+10\cdot n$ is an odd number $(\text{odd} – \text{even} + \text{even} = \text{odd})$
- $2\cdot n^5 +12 =2\cdot (n^5+6)$ is an even number (due to the even factor of 2)
- $8\cdot n^6$ is an even number (due to the even coefficient of 8)
However, the absolute parity of the expressions $11\cdot n^4$ and $n\cdot 7^3$ cannot be determined without knowing the value of $n$. In these instances, the result is odd if $n$ is odd, and even if $n$ is even.