Question

Suppose that $a$ and $b$ are digits, not both nine and not both zero, and the repeating decimal $0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible?

$3$

$4$

$5$

$8$

$9$

Answer: C

Concepts:repeating decimal factor

Difficulty rating: 1630

Solution:

Since $0.\overline{ab} = \dfrac{\overline{ab}}{99},$ the reduced denominator divides $99 = 3^2\cdot 11.$ The divisors are $1, 3, 9, 11, 33, 99.$

The denominator $1$ would require $\overline{ab} = 99,$ i.e. $a = b = 9,$ which is excluded. Each is achievable: numerators $33, 11, 9, 3,$ and $1$ reduce to denominators $3, 9, 11, 33,$ and $99,$ respectively. Thus there are $5$ possible denominators.

Thus, the correct answer is C.

2002 AMC 12A Problem 20

Answer: C

Difficulty rating: 1630

Solution:

Problem 20 in Other Years