Divisibility

Some of my children and their friends liked to solve mathematically related puzzles, and I was frequently asked to supply some. (I was at the time the author of a monthly puzzle column in the PS, the newsletter of the Princeton Section of the IEEE, and hence a reasonably good source. One day, after providing what seemed like a dozen puzzles in hardly more than a dozen minutes, I thought to get rid of them by asking a question first proposed by my predecessor puzzle author, but printed without an answer - he couldn't find one! I had provided one in time for the next edition, but not satisfactorily. (I had an answer, but not a systematic way to find one.)

This was his question: Display a number that consists of exactly one instance of each digit, and is divisible by each of the integers 1 through 10.

A computer search solves this problem easily. Anyone who actually does a systematic search will quickly find the smallest, 1234759680; several thousand solutions must exist. There is no such number that is divisible by the integers 1 through 19, but there are four if one stops at 18. Interestingly, the largest of those (4,876,391,520) is twice the smallest (2,438,195,760).

The question I actually posed to the fourth graders was more difficult: find a way to construct such a number systematically, without simply searching and testing. (I had recently devised a rather deep way that I knew was beyond them.) In two days, they had a solution that was so simple and elegant that I published the problem for the second time, alluding to their work. I got several interesting answers from members and from the elementary-school children of members. I wrote up a collage of the slickest way to express the ideas, but all solutions were essentially identical.

Return Home