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. Then, the different denominators are possible, are 

(a) 3 

(b) 4 

(c) 6 

(d) 5

Answer 1

Correct option is (d) 5

The repeating decimal \(0.\overline{ab}\) is equal to 

x = 0.abababab                    .....(i) 

100x = ab⋅abababab            .....(ii)

On subtracting Eq. (i) from Eq. (ii), we get 

99x = ab

When expressed in lowest term, the denominator of this fraction will always be a divisor of 99 = 3⋅3⋅11   

This gives us the possibilities {1, 3, 9, 11, 33, 99}. 

As a and b both are not both 9 and not both zero the denominator 1 cannot be possible.

∴ Possible denominators are {3, 9, 11, 33, 99}.