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This problem trains for: AMC-8, AMC-10, AMC-12, AIME.

What is the difference between the smallest even number and the smallest odd number that have 18 distinct positive divisors?

98 can be written as any of the following products:

Therefore, the number we are looking for may have any of the following prime factorizations:

The smallest numbers with such factorizations are the ones with the smallest prime factors at the highest exponents and larger primes at smaller powers. Also, one the numbers must be even, the other must be odd.

Out of the candidates for the even number:

it is quite obvious that the first one is the smallest.

Similarly, for the odd candidates:

the smallest one is the first.

The difference that is asked for is: