How many multiples of 3 are there from 1 to 100 which are not multiples of 2?

Concept

To count integers in a range that satisfy one divisibility rule but break another, use complementary counting: first count how many satisfy the first rule, then subtract those that also satisfy the second. A number divisible by both 3 and 2 is exactly a number divisible by their least common multiple, 6. The count of multiples of a number k in 1 to N is the integer part (floor) of N divided by k.

Application

1. Count the multiples of 3 in 1 to 100: floor of 100 divided by 3 = 33.

2. Among those, the ones that are also even are the multiples of 6 (since the lcm of 2 and 3 is 6). Count them: floor of 100 divided by 6 = 16.

3. Subtract the even ones from all multiples of 3 to keep only the ones not divisible by 2: 33 minus 16 = 17.

Cross-check

List the odd multiples of 3 directly: 3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99. Counting them gives 17, which matches. Equivalently, odd multiples of 3 are 3 times an odd number; the odd numbers from 1 to 33 are 1, 3, 5, ..., 33, and there are 17 of them.