NT-B2-M11-P009 The Period of a Fraction

Compare the remainder after \(h\) steps of long division with the initial remainder.

In long division, each new remainder is obtained by multiplying the previous one by \(10\) modulo \(n\). After \(h\) steps, the initial remainder \(a\) becomes congruent to \(a10^h\pmod n\). A period of length \(h\) means returning to the same remainder: \(a10^h\equiv a\pmod n\). Since \((a,n)=1\), we may cancel \(a\), obtaining \(10^h\equiv1\pmod n\). The least such \(h\) is the period length.

The student should understand: the period digits depend on \(a\), but the period length of a reduced fraction depends on the denominator.