An integer $n \geqslant 2$ having exactly $s$ positive divisors $1=d_{1}<d_{2}<\cdots<d_{s}=n$ is said to be good if there exists an integer $k$, with $2 \leqslant k \leqslant s$, such that $d_{k}>1+d_{1}+\cdots+d_{k-1}$. An integer $n \geqslant 2$ is said to be bad if it is not good. (a) Show that there are infinitely many bad integers. (b) Prove that, among any seven consecutive integers all greater than 2 , there are always at least four good integers. (c) Show that there are infinitely many sequences of seven consecutive good integers. (Gerhard Woeginger, Luxembourg)