How many integers between 2 and 100, inclusive, are not divisible by any odd integer greater than 1?

Rationale

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

These integers are the even numbers that are powers of 2, specifically 2, 4, 8, 16, 32, 64. Among these, the only ones that fall within the range of 2 to 100 are 2, 4, 8, 16, and 32, totaling five integers.

A (2) YOUR ANSWER

This choice incorrectly suggests that only two integers meet the criteria. However, the integers 2, 4, 8, 16, and 32 all remain undivided by any odd integer greater than 1, exceeding the count of two.

B (3) YOUR ANSWER

Choosing three integers fails to account for additional valid integers within the specified range. The integers 2, 4, 8, 16, and 32 clearly indicate that there are more than three integers that are not divisible by any odd integer greater than 1.

C (4) YOUR ANSWER

The choice of four integers is also inaccurate, as it overlooks the fifth integer, 32, which is included in the valid set of numbers. Thus, this option does not fully encompass the correct count based on the provided criteria.

E (6) YOUR ANSWER

This choice incorrectly posits that there are six integers in the range satisfying the conditions. However, the only even powers of 2 up to 100 are 2, 4, 8, 16, and 32, which total exactly five integers, not six.

Conclusion

To summarize, the integers between 2 and 100 that are not divisible by any odd integer greater than 1 are exclusively the even powers of 2: 2, 4, 8, 16, and 32. This leads to a total of five valid integers, making option D the correct answer.

How many integers between 2 and 100, inclusive, are not divisible by any odd integer greater than 1?

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

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