JEE Main · 2025 · 29 Jan 2025 · Shift 1 · Binomial Theorem

The general term of $(\sqrt[3]{7}+\sqrt[12]{11}{)}^n$ is


For this term to be an integer, $r$ must be a multiple of $12$ (making the second exponent an integer); since $12$ is itself a multiple of $3$, $n-r$ is then a multiple of $3$ precisely when $n$ is a multiple of $3$.

The valid values of $r$ are $0,12,24,\dots$, up to $n$; there are $\lfloor \frac{n}{12}\rfloor +1$ of them. Setting this to $183$:


The least $n$ in this range that is also a multiple of $3$ is $2184$ (since $2184=3\times 728$).