Zero is even or odd

Closed 7 years ago. I know that $\mathbb$ is a group under addition with a multiplication defined. I have just the definition of even and odd integers: $n$ is even if $n = 2k$ for some integer $k$ and $n$ is odd if $n = 2k+1$ for some integer $k$. Using just this I am wondering how to prove that all integers are either even or odd. That is, how can I prove that given an integer $n$, $n$ must be even or odd? My problem with this is that it seems so simple. I know that one can divide an integer by $2$ and the remainder will be $0$ or $1$. Using this, it is clear that the even and odd integers make up everything. But how can I prove it without using this fact about remainders and such? I guess one could also use facts about prime numbers, but I am looking for a proof that just uses the definition of odd and even. If you really want to avoid using general facts about division with remainder, you can use mathematical induction on $n$ to prove it for nonnegative integers: Base case: $0$ is even because $0=2\cdot 0$. Induction step: Assume that $n$ is odd or even; then we must prove that $n+1$ is also either odd or even. First subcase: $n$ is even, so $n=2k$ for some $k$. Then $n+1=2k+1$ and so it is by definition odd. Second subcase: $n$ is odd, so $n=2k+1$ for some $k$. Then $n+1=2k+1+1=2k+2=2(k+1)$, and so $n+1$ is even. That completes the induction proof, and now we just need to know that negative integers are also all either odd or even. But if $n$ is negative, then $-n$ is ...

This article describes the formula syntax and usage of the ISEVEN function in Microsoft Excel. Description Returns TRUE if number is even, or FALSE if number is odd. Syntax ISEVEN(number) The ISEVEN function syntax has the following arguments: • Number Required. The value to test. If number is not an integer, it is truncated. Remarks If number is nonnumeric, ISEVEN returns the #VALUE! error value. Example Copy the example data in the following table, and paste it in cell A1 of a new Excel worksheet. For formulas to show results, select them, press F2, and then press Enter. If you need to, you can adjust the column widths to see all the data. Formula Description Result =ISEVEN(-1) Tests whether -1 is even FALSE =ISEVEN(2.5) Checks whether 2.5 is even. The decimal portion, .5, is truncated, so 2 is tested. TRUE =ISEVEN(5) Tests whether 5 is even. FALSE =ISEVEN(0) Zero (0) is considered even. TRUE 12/23/2011 Tests the date in A6. The decimal representation of 12/23/2011 is 40900. TRUE

Example: −3, 1, 7 and 35 are all odd numbers Odd numbers are in between the even numbers. Adding and Subtracting When we add (or subtract) odd or even numbers the results are always: Operation Result Example ( red is odd, blue is even) Even + Even Even 2 + 4 = 6 Even + Odd Odd 6 + 3 = 9 Odd + Even Odd 5 + 12 = 17 Odd + Odd Even 3 + 5 = 8 (The same thing happens when we subtract instead of adding.) Multiplying When we multiply odd or even numbers the results are always: Operation Result Example ( red is odd, blue is even) Even × Even Even 4× 8 = 32 Even × Odd Even 4× 7 = 28 Odd × Even Even 5× 8 = 40 Odd × Odd Odd 5× 7 = 35 Odd Fact Every odd number has an "e" in it!

For that, we can try all the axioms formulated for even numbers. I'll use only four in this case. Note: In this question, for the sake of my laziness, I will often use $N_e$ for even, and $N_o$ for odd. Test 1: An even number is always divisible by $2$. We know that if $x,y\in \mathbb$ Passed this test too! Test 4: If $n$ is an integer of parity $P$, then $n - 2$ will also be an integer of parity $P$. We know that $2$ is even, so $2 - 2$ or $0$ is also even. Yes, the classification of naturals by their parity (= remainder modulo $2)$ extends naturally to all integers: even integers are those integers divisible by $ 2,\,$ i.e. $\rm\: n = 2k\equiv 0\pmod$. Compare the conciseness of this test to the messy reformulation that would result if we had to restrict it to positive integers. Then we could no longer represent polynomial equations in the normal form $\rm\:f(x) = 0\:$ but, rather, we would need to consider general equalities $\rm\:f(x) = g(x)\:$ where both polynomials have positive coefficients. Now the test would be much messier - bifurcating into motley cases. Indeed, historically, before the acceptance of negative integers and zero, the formula for the solution of a quadratic equation was stated in such an analogous obfuscated way - involving many cases. But by extending the naturals to the ring of integers we are able to unify what were previously motley separate cases into a single universal method of solving a general quadratic equation. Analogous examples exist t...

• • • • • • • • Related Calculators: • • • Odd or Even Calculator:If you ever need any assistance on finding even or odd numbers this tool will be of great help. You need not struggle anymore as the instant and easy to use tool will give you a detailed explanation on whether the number is even or odd. Furthermore, you will have an insight into basic details like What are Odd and Even Numbers, How to Check whether a number is odd or even, etc. Even Numbers are those that usually end with 0,2, 4, 6, 8, and odd numbers are those that end with 1, 3, 5, 7, 9. Check out the one's place firstly and you will understand whether the entire number is even or odd. To know whether a number is even or odd simply divide the number 2 and if it is exactly divisible and leaves no remainder it is even. On the Other hand, if the number isn't exactly divisible and leaves a remainder 1 it is an odd number. Examples: 20, 44, 56, 84, 78 are even numbers that leave no remainder when divided by 2. 13, 15, 79,97, 121 are odd numbers that leave remainder 1 when divided by 2. 1. What is an Even number? A number, when divided by 2, divides exactly and leaves no remainder is called even number. 2. What is an Odd Number? A number when divided by 2 leaves a remainder 1 is called Odd Number 3. Is 122 an Even Number? Given Number is 122 On dividing the number 122 by 2 you will have a zero remainder. As the number 122 is exactly divisible by 2 it is an even number. 4. Where do I get detailed steps on how to ...