Let $j$ be John number and $m$ be Mary number. Then $2002=j m$ or $2002=j+m$ Behind John answer, Assume that $2002$ is not a multiple of $j$ then $2002=j+m$ so John should guess the number. So $2002$ is a multiple of $j$. Behind Mary answer, Similarly $2002$ is a multiple of $m$ Assume that $2002-m$ is not a divisor of $2002$ then $2002=jm$ so Mary should guess the number. So $2002-m$ is a divisor of $2002$ Conclusion So $2002-m$ and $m$ are both divisors of $2002$ only for $m=1001$ and $j=2$