2013 UNCO Math Contest II Problems/Problem 8

Problem

EXAMPLE: The non-terminating periodic decimal $0.124124 \ldots = 0.\overline{124}$ has period three and is abbreviated by placing a bar over the shortest repeating block.

(a) If all digits $0$ through $9$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2 \ldots d_6}$ have period exactly six? Do not include patterns like $0.323$ and $0.17$ that have shorter periods.

(b) If only digits $0$ and $1$ are allowed, how many distinct periodic decimals $0.\overline{d_1d_2\ldots d_{12}}$ have period exactly $12$?


Solution

(a) $998,910$ (b) $4,020$

See Also

2013 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 7
Followed by
Problem 9
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions