2021 JMC 10 Problems/Problem 8
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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.