2. [2] Suppose $N$ is a 6-digit number having base-10 representation $\underline{a} \underline{b} \underline{c} \underline{d} \underline{e} \underline{f}$. If $N$ is $6 / 7$ of the number having base-10 representation $\underline{d} \underline{e} \underline{f} \underline{a} \underline{b} \underline{c}$, find $N$.
Answer: 461538
Solution: We have $7(a b c d e f)_{10}=6(\text { de fabc })_{10}$, so $699400 a+69940 b+6994 c=599300 d+59930 e+$ $5993 f$. We can factor this equation as $6994(100 a+10 b+c)=5993(100 d+10 e+f)$, which yields $538(a b c)_{10}=461(\text { def })_{10}$. Since $\operatorname{gcd}(538,461)=1$, we must have $(a b c)_{10}=461$ and $(\text { def })_{10}=538$.