# Questions tagged [parity]

This tag is for questions relating to "Parity", a mathematical term that describes the property of an integer's inclusion in one of two categories: even or odd.

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### A question regarding cyclic codes [closed]

Let $C$ be the smallest cyclic code over $F_3$, which contains the word $11020201$. A.What is the generating polynomial $g(x)$ of the code $C$ ? B. Write the generating matrix and the parity check ...
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### Is it possible to determine if $\frac{xy^2}{2}$ is an even number?

The problem Given $x, y \in \mathbb{Z}$, is it possible to determine if $\frac{xy^2}{2}$ is an even number? $x$ and $y$ are consecutive numbers and $x$ is even. My attempt Assuming $n$ is an integer, ...
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### Function that maps any given positive even number and its subsequent (odd) integer to the same value? [closed]

I’m looking for a function $f$ such that $f(2n)$ is uniquely equal to $f(2n+1), n \in \mathbb{Z}^+$. The only one I came up with is the floor function, i.e. $f(x)= \lfloor{\frac{x}{2}}\rfloor$, but ...
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### A proof of the theorem $1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ with $k$ odd and $m$ even

I tried to prove the following theorem : If $n>1$ then $\displaystyle1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ where $k$ is odd and $m$ is even. and I'd like to know if there is any flaw in ...
Prove that any positive integer with exactly $1000$ divisors, which when arranged in increasing order have alternating parity (so the first divisor is odd, the second is even, the third is odd, etc. ...