3. Determine a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, gives six-digit numbers written with the same digits as the original number.
Solution. From the analysis of the products with 2, 3, 4, 5, and 6, it is derived that the only solution to the problem is the number 142857. For example, if the desired number is $\overline{a b c d e f}$, then since multiplying by the number 6 should yield a six-digit number written with the same digits as the original number, we conclude that $a=1$, etc. We leave the details to the reader for practice.