2021 JMC 10 Problems/Problem 8
Problem
A positive integer is pretentious if it has both even and odd digits. For example, 
and 
are pretentious. How many pretentious three-digit numbers are odd?
Solution
Suppose an odd pretentious three-digit number is of the form 
where 
equals 
or 
Both 
and 
can be even, be even and odd, odd and even, but cannot both be odd. Using complementary counting, there are 
total choices for 
and 
undesired cases (when both digits are odd), leaving 
desired pairs . Thus, the answer is 
such numbers.
