The largest $a$ such that $7^{a} \left\lvert\,\binom{ 2 k}{k}\right.$ is equal to the number of carries when you add $k+k$ in base 7 , by Kummer's Theorem. Thus, we need two carries, so $2 k$ must have at least 3 digits in base 7 . Hence, $k \geq 25$. We know $k=25$ works because $25+25=34_{7}+34_{7}=101_{7}$ has two carries.