To find the number of integers between 1 and 1000 that are not divisible by either 11 or 35, we can use the principle of inclusion-exclusion.
Step 1: Find the total number of integers between 1 and 1000.
There are 1000 integers between 1 and 1000, inclusive.
Step 2: Find the number of integers divisible by 11.
11, 22, 33, ..., 990 (91 integers)
Step 3: Find the number of integers divisible by 35.
35, 70, 105, ..., 980 (28 integers)
Step 4: Find the number of integers divisible by both 11 and 35 (their least common multiple, which is 385).
385, 770 (2 integers)
Step 5: Apply the principle of inclusion-exclusion.
Total integers - (integers divisible by 11 + integers divisible by 35 - integers divisible by both)
= 1000 - (91 + 28 - 2)
= 1000 - 117
= 883
However, this calculation excludes the integers 1 and 1000, which are not divisible by either 11 or 35. Therefore, we need to add 2 to the result.
883 + 2 = 885
But this is not among the answer choices. Let’s recheck our calculation.
Step 2: Find the number of integers divisible by 11.
11, 22, 33, ..., 990 (90 integers, not 91)
Step 3: Find the number of integers divisible by 35.
35, 70, 105, ..., 980 (28 integers)
Step 4: Find the number of integers divisible by both 11 and 35 (their least common multiple, which is 385).
385, 770 (2 integers)
Step 5: Apply the principle of inclusion-exclusion.
Total integers - (integers divisible by 11 + integers divisible by 35 - integers divisible by both)
= 1000 - (90 + 28 - 2)
= 1000 - 116
= 884
The correct answer is:
A. 884