Sunday Times Teaser 2596 – Unusual Factors
by Nick MacKinnon
Published: 24 June 2012 (link)
There are no neat tests for divisibility by 7 or 13, so it’s unusual for these factors to feature in Teasers: today we put that right. If you start with any three different non-zero digits, then you can make six different three- figure numbers using precisely those digits. For example, with 2, 5 and 7 you get 257, 275, 527, 572, 725 and 752. Here, one of their differences (572 — 257) is divisible by 7, and another (725 — 257) is divisible by 13. Your task today is to find three different digits so that none of the six possible three- figure numbers and none of the differences between any pair of them is a multiple of either 7 or 13.
What are the three digits?