The numbers between 10 and 30 that have exactly 5 factors are 16 and 25. The number 16 has factors 1, 2, 4, 8, and 16. The number 25 has factors 1, 5, 25, but since it is a perfect square, it also includes the repeated factor of 5. These are the only two numbers between 10 and 30 that have exactly 5 factors.
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A number that has exactly an odd number of factors is a power of another number. A number between 10 and 30 that is a power is 16. It is 24 (as well as 42) and its factors are 1, 2, 4, 8, and 16.