Question 13

There are 5 integers between 2 and 100 that are not divisible by any odd integer greater than 1.

The integers that meet this criterion are the powers of 2, specifically 2, 4, 8, 16, and 32. These numbers are not divisible by any odd integers greater than 1, as they are solely products of the prime factor 2.

A) 2

This option is incorrect because only listing the number 2 does not account for the additional integers that also fit the criteria. Besides 2, there are several other powers of 2 within the specified range.

B) 3

This option is also incorrect. While 2, 4, and 8 might come to mind when considering the numbers not divisible by odd integers greater than 1, it excludes several other qualifying integers, particularly those that are higher powers of 2.

C) 4

Option C is not correct because, although it includes 2 and 4, it fails to include all valid integers up to 100 that meet the criteria. The integers 8, 16, and 32 are also valid and should be counted.

D) 5

This option is correct as it includes the integers 2, 4, 8, 16, and 32, which are the only integers between 2 and 100 that are not divisible by any odd integer greater than 1. Each of these numbers is a power of 2.

E) 6

This option is incorrect because it suggests that there are more than five qualifying integers. The only integers that satisfy the condition remain limited to 2, 4, 8, 16, and 32, thus making six an incorrect count.

Conclusion

The correct answer is 5, as it accurately represents the integers between 2 and 100 that are solely powers of 2. All other options fail to account for the complete set of integers that fit the specified criteria, leading to an undercount of valid numbers. Therefore, option D definitively encapsulates the correct total.