Problem:
How many factors of have more than factors? (As an example, has factors, namely and .)?
Answer Choices:
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Solution:
The number 2020 has 12 factors.
These factors may be classified as follows:
The number has exactly factor.
The numbers , and are primes, so each has exactly factors.
The number has exactly factors, namely , and .
The remaining numbers, , and , each have more than factors.
Thus of the factors of have more than factors.
In order to determine the number of factors of , note that the prime factorization of is
This means that each factor of has the form , where , or ; or ; and or . Thus the number of factors of is
By similar reasoning, any positive integer with at least distinct prime factors, say and , has at least factors, namely and .
The factors of that have no more than distinct prime factor are and . Each of these has at most factors. The remaining factors, and , each have at least distinct prime factors. Therefore of the factors of have more than factors.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions