Problem of the Week 1093

An Alphanumeric Puzzle

Let w represent an n-letter word containing at most 10 different letters, like KONHAUSER, PROBLEMFEST, or PROBLEMOFWEEK.

True or False: Any such w can be made divisible by 7 under some assignment of the 10 decimal digits to the letters of w (different letters replaced by different digits, same letters replaced by same digits).

Hard Extra Credit: Call an integer d "attainable" if for any sufficiently long word w as above there is a digit substitution so that the resulting decimal integer is divisible by d. Find all "attainable" integers. Note: "Sufficiently long" is needed since, for example, the word AB will never be divisible by 541, but it is conceivable that 541 is nevertheless attainable.

Source: N. Kildonan, Problem 1859, Crux Mathematicorum, 20:6, June 1994, 168-170.

© Copyright 2008 Andrew Beveridge and Stan Wagon. Reproduced with permission.