Problem
NT-B2-M11-P011 A Fraction Repeating Its Denominator
#11
★★★☆☆ Level 3 of 5
Find all positive integers \(n\) for which the decimal fraction \(\frac1n\) equals \(0.\overline{n}\), where the repeating block is the decimal notation of \(n\) itself.
1001 Problems in Classical Number Theory (method inspiration) · Problem 262
If \(n\) has \(k\) digits, then \(0.\overline{n}=\frac{n}{10^k-1}\).
Let \(n\) have \(k\) digits. Then \(0.\overline{n}=\frac{n}{10^k-1}\), so the condition gives \(\frac1n=\frac{n}{10^k-1}\), hence \(n^2=10^k-1\).
If \(k=1\), then \(n^2=9\), so \(n=3\). If \(k\ge2\), then \(10^k\) is divisible by \(4\), so \(10^k-1\equiv3\pmod4\), but no square is congruent to \(3\pmod4\). Thus there are no other solutions. The answer is \(n=3\).
A good example of converting a repeating decimal into a fraction.