Question

How many integers from 1 to 1000 are multiples of 6 or multiples of 8

Answer 1

There are 250 integers between 1 to 1000 that are multiples of 6 or multiples of 8, after calculating the multiples of each and subtracting the multiples of 24, which is their least common multiple.

Step-by-step explanation:

To find how many integers from 1 to 1000 are multiples of 6 or multiples of 8, we need to count the multiples of each and then subtract the multiples of their least common multiple (LCM) to avoid double counting. The LCM of 6 and 8 is 24.

First, the number of multiples of 6 between 1 and 1000 is calculated by dividing 1000 by 6, which gives us 166 with a remainder. So, there are 166 multiples of 6.

Next, the number of multiples of 8 between 1 and 1000 is calculated by dividing 1000 by 8, which gives 125 exact multiples of 8.

For the LCM of 24, we divide 1000 by 24, which yields 41 with a remainder. Therefore, there are 41 multiples of 24 within that range.

To find the final count, we add the multiples of 6 and the multiples of 8, then subtract the multiples of 24 to correct for any numbers that have been counted twice.

The calculation then is 166 + 125 - 41, resulting in 250. That means there are 250 integers between 1 to 1000 that are multiples of 6 or multiples of 8.

Answer 2

To find the number of integers that are multiples of 6 or multiples of 8 from 1 to 1000, we can use the principle of inclusion-exclusion.

Step-by-step explanation:

To find the number of integers from 1 to 1000 that are multiples of 6 or multiples of 8, we can use the principle of inclusion-exclusion. First, we find the number of multiples of 6, which is 1000/6 = 166. Then, we find the number of multiples of 8, which is 1000/8 = 125. However, there are some integers that are multiples of both 6 and 8, so we need to subtract the number of multiples of their least common multiple, which is 24. Therefore, the number of integers that are multiples of 6 or multiples of 8 is 166 + 125 - 24 = 267.