Question

Let Z be the set of all integers. Let X = 1 ≤ n ≤ 1000.

How many integers in X are divisible by 4 or 6 or 15?

Answer 1

To find how many integers in the set X are divisible by 4, 6, or 15, one must use the inclusion-exclusion principle. Count the integers divisible by each, then consider the overlaps by using LCMs of the numbers, and make necessary subtractions and additions. The final count reveals that there are 366 integers within the set X that satisfy the given conditions.

Step-by-step explanation:

To determine how many integers in the set X = 1 ≤ n ≤ 1000 are divisible by 4, 6, or 15, we must apply the principle of inclusion-exclusion. First, we find the count of integers divisible by each of these numbers within the given range.

The number of integers divisible by 4 is found by dividing 1000 by 4, yielding 250.
The number of integers divisible by 6 is found by dividing 1000 by 6, yielding around 166.66, which we round down to 166, since we're counting whole numbers only.
The number of integers divisible by 15 is found by dividing 1000 by 15, yielding about 66.66, which we round down to 66.

Next, we find the count of integers divisible by the least common multiples of these numbers in pairings:
The LCM of 4 and 6 is 12, and there are 1000/12 = 83 integers divisible by both.
The LCM of 4 and 15 is 60, and there are 1000/60 = 16 integers divisible by both.
The LCM of 6 and 15 is 30, and there are 1000/30 = 33 integers divisible by both.

However, some integers are divisible by all three numbers (4, 6, and 15), which means they are divisible by the LCM of these three numbers. The LCM of 4, 6, and 15 is 60, which we have already found to be 16.

Applying the inclusion-exclusion principle, we calculate the total number as follows:

1. Add up the numbers of divisible integers: 250 (by 4) + 166 (by 6) + 66 (by 15) = 482.
2. Subtract the numbers divisible by each pair of LCMs: 482 - 83 (by 12) - 16 (by 60, already subtracted, since it's the LCM of all three) - 33 (by 30) = 350.
3. Add back the numbers divisible by the LCM of all three: 350 + 16 = 366.

Therefore, there are 366 integers in the set X that are divisible by 4, 6, or 15.