Pick a random pair $ (a,b)\in\mathbb Z_n^2\backslash\{0,0\}$ . Denote $ N_2(a,b,n)$ to be minimum $ \ell_r$ norm of vector $ (x,y)$ as $ (x,y)$ ranges over all nonzero integral solutions to $ (x,y)\equiv t(a,b)\bmod n$ where $ t\in\mathbb Z$ with $ 0<t<n$ .
Now let integers $ a,b$ be of size in $ (\sqrt m,m\sqrt m)$ .
Let $ p_1,p_2$ be primes of similar size some integer $ m$ and let $ t_1$ attain $ N_2(a,b,p_1)$ and $ t_2$ attain $ N_2(a,b,p_2)$ . From Dirichlet pigeonhole we know $ N_2(a,b,p_1)$ and $ N_2(a,b,p_2)$ are less than $ \sqrt{2m}$ .
Let $ t$ attain $ N_2(a,b,p_1p_2)$ (note $ (t,p_1p_2)=1$ need not hold while $ (t_1,p_1)=(t_2,p_2)=1$ holds).
Then $ N_2(a,b,p_1p_2)$ seems to be typically $ m\sqrt 2$ .

Is it possible for $ N_2(a,b,p_1p_2)$ to be smaller than $ \sqrt{2m}\mu$ if $ N_2(a,b,p_1)$ and $ N_2(a,b,p_2)$ are greater than $ \sqrt{2m}\mu $ at a $ \mu\in(0,\sqrt{2m})$ ?

If so what is the probability?
This is the result from Akshay Venkatesh when $ n$ is prime. Then it is true as $ n\rightarrow\infty$ the distribution of $ N_2(a,b,n)/\sqrt{n}$ coincides with distribution of $ 1/\sqrt y$ where $ x+iy$ is picked at random with respect to hyperbolic measure from $ \{z:z\geq1,\mathcal R(z)\leq\frac12\}$ .