Cyclic pairs

When you transpose the digits of a two digit number and obtain a multiple of the original, the original may be called a “cyclic pair”. I have before me, a computer printout obtained by Mr F. B. Satterthwaite of 439 East 51st Street, New York 10022, exhibiting all the cyclic pairs for which the multiple of the original obtained by transposing the digits is ⩽ 16. This computer printout exhibits patterns of many kinds. The most striking pattern that Satterthwaite identified was that cyclic pairs do not exist when the base is one less than a prime number, but do exist for every other base ⩾ 5. His claim may be substantiated as follows.