How many positive integers not exceeding 100 are divisible by 3 or 11?

Step 1: Find the number of integers divisible by 3

The smallest positive integer divisible by 3 is 3 and the largest positive integer not exceeding 100 that is divisible by 3 is 99.

To find the total number of such integers, we use the formula for the nth term of an arithmetic sequence, which is a_n = a_1 + (n - 1) * d, where a_n is the nth term, a_1 is the first term, and d is the common difference.

In this case, a_1 is 3, a_n is 99, and d is 3.

So, we have 99 = 3 + (n - 1) * 3.

Solving for n, we get n = 33.

So, there are 33 positive integers not exceeding 100 that are divisible by 3.

Step 2: Find the number of integers divisible by 11

The smallest positive integer divisible by 11 is 11 and the largest positive integer not exceeding 100 that is divisible by 11 is 99.

Using the same formula as above, with a_1 = 11, a_n = 99, and d = 11, we get 99 = 11 + (n - 1) * 11.

Solving for n, we get n = 9.

So, there are 9 positive integers not exceeding 100 that are divisible by 11.

Step 3: Find the number of integers divisible by both 3 and 11

The least common multiple of 3 and 11 is 33.

The smallest positive integer divisible by 33 is 33 and the largest positive integer not exceeding 100 that is divisible by 33 is 99.

Using the same formula as above, with a_1 = 33, a_n = 99, and d = 33, we get 99 = 33 + (n - 1) * 33.

Solving for n, we get n = 3.

So, there are 3 positive integers not exceeding 100 that are divisible by both 3 and 11.

Step 4: Subtract the number of integers divisible by both from the sum of the number of integers divisible by each

The total number of positive integers not exceeding 100 that are divisible by 3 or 11 is 33 + 9 - 3 = 39.