Final answer:
To find the number of integers from 1 through 1,000 that are multiples of 2 or multiples of 9, we can divide 1,000 by each number and count the integers. However, we need to correct for counting the integers twice if they are multiples of both 2 and 9. The total number of integers is 556.
Step-by-step explanation:
We need to find the number of integers from 1 through 1,000 that are multiples of 2 or multiples of 9. To find the multiples of 2, we can divide 1,000 by 2 which gives us 500 integers. To find the multiples of 9, we can divide 1,000 by 9 which gives us 111 integers. However, some integers may be counted twice because they are multiples of both 2 and 9.
To correct this, we need to find the common multiples of 2 and 9, which are multiples of their least common multiple, which is 18. To find these multiples, we can divide 1,000 by 18 which gives us 55 integers. So, the total number of integers that are multiples of 2 or multiples of 9 is 500 + 111 - 55 = 556.