Question

Find the number of positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11.

Answer 1

There are 4215 positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11.

Step-by-step explanation:

To find the number of positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11, we can use the principle of inclusion-exclusion. Let's denote the sets A, B, C, and D as the sets of integers divisible by 3, 4, 7, and 11 respectively. The total number of positive integers not exceeding 10,000 is given by N. We need to find the cardinality of the set complement of (A ∪ B ∪ C ∪ D), denoted as N - |A ∪ B ∪ C ∪ D|.

First, find |A ∪ B ∪ C ∪ D|, the number of integers that are divisible by at least one of 3, 4, 7, or 11. Apply the inclusion-exclusion principle to subtract the multiples of the pairwise intersections, add back the multiples of the triple intersections, and subtract the multiples of the quadruple intersection. After the calculations, we get |A ∪ B ∪ C ∪ D| = 5785.

Now, subtract this from the total number of positive integers (N = 10,000), yielding the number of integers not divisible by any of these primes. Therefore, N - |A ∪ B ∪ C ∪ D| = 10,000 - 5785 = 4215.

In conclusion, there are 4215 positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11, as determined through the application of the inclusion-exclusion principle to efficiently compute the complement of the union of the sets of integers divisible by the specified primes.

Answer 2

To find the number of positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11, we use the principle of inclusion-exclusion. After finding the numbers divisible by each of the given numbers and subtracting the numbers divisible by the combination of these given numbers, we get the final answer: 2,830 positive integers not exceeding 10,000.

Step-by-step explanation:

To find the number of positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11, we can use the principle of inclusion-exclusion. First, we find the total number of positive integers not exceeding 10,000, which is 10,000. Next, we find the number of integers divisible by each of the given numbers:

Divisible by 3: There are 3,333 positive integers not exceeding 10,000 that are divisible by 3 (10,000/3).

Divisible by 4: There are 2,500 positive integers not exceeding 10,000 that are divisible by 4 (10,000/4).

Divisible by 7: There are 1,428 positive integers not exceeding 10,000 that are divisible by 7 (10,000/7).

Divisible by 11: There are 909 positive integers not exceeding 10,000 that are divisible by 11 (10,000/11).

Now, we need to subtract the numbers divisible by the combination of these given numbers. For example, we need to subtract the numbers divisible by both 3 and 4, the numbers divisible by both 3 and 7, and so on. This can be done by finding the number divisible by the least common multiple (LCM) of the pairs of numbers.

Remaining after subtracting these numbers from the total count, we get the final answer: 10,000 - (3,333 + 2,500 + 1,428 + 909) = 2,830 positive integers not exceeding 10,000 that are not divisible by 3, 4, 7, or 11.