How many divers long could the repeating decimal be if the denominator of the rational numbers is 7?

A.Up to six digits long
B.Any future length
C.Up to 7 digits long
D.Infinitely long

1 answer

The length of the repeating decimal for a rational number depends on the denominator when the fraction is in its simplest form. For a denominator of 7, the repeating decimal can have a maximum period of \( n \), where \( n \) is the smallest integer such that \( 10^n \equiv 1 \mod 7 \).

Calculating this:

  • \( 10^1 \equiv 3 \mod 7 \)
  • \( 10^2 \equiv 2 \mod 7 \)
  • \( 10^3 \equiv 6 \mod 7 \)
  • \( 10^4 \equiv 4 \mod 7 \)
  • \( 10^5 \equiv 5 \mod 7 \)
  • \( 10^6 \equiv 1 \mod 7 \)

Since \( n = 6 \) is the smallest integer where the equivalence holds, the repeating decimal can be up to 6 digits long.

Thus, the correct answer is:

A. Up to six digits long.