Is $\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+1)\})=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+3)\})$?

Question

Let $\alpha\ge 1$ be an even integer, and $k$ an intger s.t, $1\le k\le \alpha$. Set $\alpha'=\alpha/2$, $$ A=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+1)\}),\quad B=\mathrm{Card}(\{n : 0\le n\le k-1, k\mid\alpha'(4n+3)\}) $$ So my question is: Can we find $A$ and $B$ explicitly in terms of $\alpha'$ and $k$? I tried with some examples and I suspect that, if $2^g\mid\mid \alpha$ (i.e., $\alpha=2^g\beta$ with $\gcd(\beta,2)=1$) then

$$ A=B=\left\{ \begin{array}{cc} 0 & \mbox{if} \;k\; \mbox{multiple of} \;2^g\\ \gcd(\alpha',k) & \mbox{otherwise.} \\ \end{array} \right. $$ I am pretty sure that this is true,but I don't know how to prove it formally.

Answer 1

This is a straightforward exercise in elementary number theory, but I felt like doing it.

For any integer $m$, the relation $k\mid\alpha'm$ is equivalent to $\frac{k}{\gcd(k,\alpha')}\mid m$. $A$ is the number of such $m$'s with the additional properties $m\equiv 1\pmod{4}$ and $1\leq m<4k+1$, while $B$ is the number of such $m$'s with the additional properties $m\equiv 3\pmod{4}$ and $3\leq m<4k+3$. As both $A$ and $B$ count odd integers, they are zero unless $\frac{k}{\gcd(k,\alpha')}$ is odd, i.e. the exponent of $2$ in $k$ is at most the exponent of $2$ in $\alpha'$, i.e. $2^g\nmid k$. Now assume that this is the case, then by the Chinese remainder theorem both $A$ and $B$ count integers $m$ that lie in a fixed residue class modulo $\frac{4k}{\gcd(k,\alpha')}$ and also in an interval of $4k$ consecutive integers. Hence both $A$ and $B$ equal $4k$ divided by $\frac{4k}{\gcd(k,\alpha')}$, which is $\gcd(k,\alpha')$.