Even and Odd Integers

Even and Odd Integers

• An even integer is a number that is divisible by two. An odd integer is a number that is not divisible by two.

• Divisibility by two can be checked by the last digit: a number ends in $0, 2, 4, 6, 8$ if and only if it is divisible by two.

• Negative integers can be even or odd as well. Examples: $-10\text{ is even}, \;-9\text{ is odd}.$

• Zero is divisible by two; zero is a multiple of two, hence it is even. (We will discuss zero more in a future video.)

• In parity language, even numbers have the form $n=2k$ for some integer $k$, and odd numbers have the form $n=2k+1$.

• Base intuition: parity is a property modulo 2.

Parity Rules for Addition and Subtraction

• If you add or subtract two even numbers, the result is even.

  • Example: $4+6=10$ and $6-4=2$ (both even).

• If you add or subtract an even and an odd number, the result is odd.

  • Examples: $10-3=7$ and $10+3=13$ (both odd).

• If you add or subtract two odd numbers, the result is even.

  • Examples: $5+3=8$ and $5-3=2$ (both even).

• Quick way to remember: write even as $2k$ and odd as $2k+1$. Then:

  • $(2k) \pm (2m) = 2(k \pm m)$ (even)

  • $(2k) \pm (2m+1) = 2(k \pm m) \pm 1$ (odd)

  • $(2k+1) \pm (2m+1) = 2(k+m+1)$ (even)

Parity Rules for Multiplication

• If you multiply two even numbers, the result is even.

  • Examples: $4\times6=24$, $4\times8=32$ (even).

• If you multiply an even number by an odd number, the result is even.

  • Examples: $4\times3=12$, $4\times5=20$. Note: negatives work the same: $4\times(-5)=-20$ (even).

• If you multiply two odd numbers, the result is odd.

  • Examples: $3\times5=15$, $3\times7=21$, $9\times3=27$.

• A helpful way to see this with algebra: if the numbers are expressed as $n=2k$ or $n=2k+1$, their products preserve parity as above.

Division: Caveats and Exceptions

• Division parity rules are not as clean as addition, subtraction, or multiplication. The quotient can be even, odd, or a non-integer (neither even nor odd as an integer).

• Even ÷ Even:

  • Example: $12\div 2 = 6$ (even).

  • Example: $14\div 2 = 7$ (odd).

  • Example: $18\div 4 = 4.5$ (neither integer, i.e., not even or odd as an integer).

• Even ÷ Odd:

  • Example: $12\div 3 = 4$ (even).

  • Example: $26\div 13 = 2$ (even), but parity is not guaranteed to be the same in all cases.

  • Example: $12\div 5 = 2.4$ (neither integer).

• Odd ÷ Odd:

  • Example: $9\div 3 = 3$ (odd).

  • Example: $21\div 3 = 7$ (odd).

  • Example: $21\div 5 = 4.2$ (neither integer).

• Takeaway: division does not have a simple parity rule like addition or multiplication; the result can be even, odd, or non-integer depending on the numbers involved. When studying parity, focus more on addition, subtraction, and multiplication.

Exponents with Even and Odd Bases

• Even base raised to an integer exponent:

  • For any even base $a=2k$ and exponent $n\in\mathbb{Z}$ with $n\neq 0$, $a^n$ is even.

  • If the exponent is zero, $a^0 = 1$ (which is odd).

  • The standard rule (as noted in this course) is: anything raised to the zero power is 1; we will explore this more in a future video about zero.

  • Examples: $2^2=4$ (even), $2^3=8$ (even), $2^0=1$ (odd).

• Odd base raised to an integer exponent:

  • For any odd base $b=2k+1$ and any nonnegative exponent, $b^n$ is odd. This includes the case $n=0$, since $b^0=1$, and $1$ is odd.

  • Examples: $3^3=27$ (odd), $3^4=81$ (odd), $3^0=1$ (odd).

• Summary:

  • If the base is even and the exponent is a positive integer, the result is even (
    except the zero-exponent case).

  • If the base is odd, the result is always odd for nonnegative exponents.

  • The special case of exponent zero yields 1, which is odd.

Quick References and Practice Prompts

• Parity definitions in compact form:

  • Even: $n=2k, k\in\mathbb{Z}$

  • Odd: $n=2k+1, k\in\mathbb{Z}$

• Divisibility by 2: last digit rule (ends with $0,2,4,6,8$).

• Important edge cases to remember:

  • Zero is even: $0=2\cdot 0$.

  • Negative numbers retain parity: e.g., $-8$ is even, $-7$ is odd.

  • For exponents, $a^0=1$ for any nonzero $a$; if the base is even, the result is even for positive exponents, and odd for exponent 0.

• Encouragement: if you forget a rule, create your own examples to test parity; practice with multiple numbers to see the patterns.

Connections to Foundations and Real-World Relevance

• Parity is a fundamental concept in modular arithmetic (mod 2). It underpins algorithms in computer science (binary logic), cryptography, and error detection.

• In many programming tasks, quick parity checks (like last digit or bitwise & operations) speed up decision-making without full arithmetic.

• Zero’s special status (even, multiple of 2, yet sometimes treated separately in certain contexts) motivates deeper study of edge cases in math.

• Real-world relevance: parity reasoning helps in quick mental math, simplifying problems in number theory, and proving divisibility properties.

Notes for Exam Preparation

• Be able to derive parity results from the standard representations $n=2k$ or $n=2k+1$.

• Practice with both positive and negative integers, and include the zero case.

• Memorize the clean rules for addition, subtraction, and multiplication; remember that division has many exceptions and is not as straightforward for parity.

• For exponents, remember:

  • If the base is even and exponent > 0, result is even.

  • If the base is odd, result is odd for any nonnegative exponent.

  • The exponent zero yields 1 for any nonzero base (and 1 is odd).

• Build your own examples to reinforce understanding and to prepare for similar questions on exams.