JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Binomial Theorem

The condition $3^n-3\equiv 0(\mathrm{mod}7)$ means $3^n\equiv 3(\mathrm{mod}7)$. The powers of $3$ modulo $7$ cycle as $3,2,6,4,5,1$ with period $6$, so $3^n\equiv 3$ exactly when $n\equiv 1(\mathrm{mod}6)$. In $10\le n\le 100$ these are $13,19,25,\dots ,97$, an arithmetic progression of common difference $6$, whose count is $\frac{97-13}{6}+1=15$. Hence the set has $15$ elements.