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Problem of the Week 1036

Rational Coincidence

How many rational numbers a/b are there (with a and b relatively prime) such that a/b = b.a where the dot is interpreted as the usual base-10 dot? That is, you seek positive integers a and b such that the base 10 number obtained by placing b and a side-by-side with a decimal point between them exactly equals a/b.

Source: Hungary-Israel Math Competition: The First Twelve Years, S. Gueron, Australian Mathematics Trust; Problem 1994.1

Extra Credit: Suppose we work in base B. So now we ask for two integers such that the rational number a/b is identical to the rational obtained as b.a in base B, meaning b + (a / B#base-B) digits in a. In base 6 we have the example 18/4 because 18 is 30 in base 6, so we form 4.30 in base 6, which is the rational 9/2, which equals 18/4. Yet this example is a little unsatisfactory because 18/4 is not in lowest terms. I did some checking by computer and since I found no other examples, let me state the following conjecture (can anyone prove it?)

Conjecture: 10 is the first base for which there exists two relatively prime integers a and b so that a/b is identical with b.a.

[View the solution]

12 September 2005