1. Given $k\in\Bbb N$ is there coprime $1<a,1<b$ with $(k,a^2-b^2)=1$ and coprime $1<c,1<d$ such that $k|(ac-bd)$ and $k|(ad-bc)$?
2. What is the smallest $\max(|a|,|b|,|c|,|d|)=\max(a,b,c,d)$ among such $a,b,c,d$?



On symmetric linear diophantine equations
From your assumptions follows that $$k\mid (ac-bd\pm(ad-bc))=(a\mp b)(c\pm d).$$ So $k\mid(c\pm d)$, i.e. $k\mid(c+d,c-d)\in\{1,2\}.$


