# 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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### If $\frac{p^2}{q^2} + \frac{r^2}{s^2} = 1$, then $q,s$ are odd and one of $p,r$ is even

Suppose $\frac{p}{q}$ and and $\frac{r}{s}$ are rationals in lowest terms (so $\gcd (p,q) = \gcd(r,s) = 1$) and $\frac{p^2}{q^2} + \frac{r^2}{s^2} = 1$; i.e. $p^2s^2+r^2q^2=q^2s^2$. Then exactly one ...
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### Proving that if Generator Matrix is in standard form then parity check matrix is a generator matrix for not C and so a parity check matrix

I have to prove that: Is $G=\left(I_{k} P\right)$ a generator matrix for a $[n, k]$ -Code $C$ then $H=\left(-P^{\top} I_{n-k}\right)$ is a generator matrix for $C^{\perp}$ and so it is a a parity ...
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### Floor function parity problem

Prove that for every natural k this expression is always odd $⌊(5+\sqrt{19})^k⌋=A^k$ Progress that I' ve done is: I noticed $9^k<A^k<(9.5)^k$ Also I tried an induction approach, I used Binomial ...
Que. If the quadratic $ax^2 + bx + c$ has a rational root, and $a$, $b,$ and $c$ are integers, then A) at least one of $a, b, c$ is even B) all of $a,b,c$ are even C) at most one of $a,b,c$ is odd D)...