Safety in numbers: Cracking the coefficient code

Decimal expansion of numbers was a common exercise in middle-school mathematics. Fractions like 1/5 and 1/4 were less taxing. Their decimal expansions terminated promptly at 0.2 and 0.25 respectively. But the fraction 1/3 was a little intriguing. Its decimal expansion did not terminate, but it soon became clear that a single digit, namely 3, started to repeat itself. For certain other fractions, a whole block of digits, and not a single one, would repeat interminably. For example, the fraction 1