Patrícia wrote, in ascending order, the positive integers formed only by odd digits: 1, $3,5,7,9,11,13,15,17,19,31,33, \ldots$ What was the $157^{\text{th}}$ number she wrote?
A) 997
B) 999
C) 1111
D) 1113
E) 1115
Solution

## ALTERNATIVE D

There are five odd digits: 1, 3, 5, 7, and 9. Counting only positive integers, there are then 5 numbers formed by only one odd digit, $5 \times 5=25$ numbers formed by two odd digits, and $5 \times 5 \times 5=125$ numbers formed by three odd digits. Thus, there are $5+25+125=155$ positive integers less than 1000 formed by odd digits. The $156^{\text{th}}$ is then 1111 and the $157^{\text{th}}$ is 1113.