= Strong-pseudoprime bases for a product of two primes with common predecessor divisor ten
Let $N=pq$ for distinct odd <prime numbers> $p,q$ with $\gcd(p-1,q-1)=10$. Exactly $100$ unit bases make $N$ a <Fermat pseudoprime>, and exactly $50$ of these make it a <strong pseudoprime>. The Fermat bases form $C_{10}\times C_{10}$ under the <Chinese remainder theorem for unit groups>. If $N-1=2^sm$, then $5\mid m$ and $m$ is odd, so the $m$th-power map sends five elements of each $C_{10}$ to $1$ and five to $-1$. The strong condition requires equal signs in both components, giving $5^2+5^2=50$.
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