Final answer:
There are 360 3-digit numbers that are divisible either by 5, 7, or 9.
Step-by-step explanation:
To determine the number of 3-digit numbers that are divisible by 5, 7, or 9, we need to count the numbers that are divisible by each of these numbers and then subtract the numbers that are divisible by more than one of these numbers.
To count the numbers divisible by 5, we need to find the number of multiples of 5 between 100 and 999. The quotient of dividing 999 by 5 is 199, and the quotient of dividing 100 by 5 is 20. So, there are 199 - 20 + 1 = 180 numbers divisible by 5.
Similarly, there are 128 numbers divisible by 7 and 111 numbers divisible by 9.
To find the numbers divisible by both 5 and 7, we need to find the multiples of the least common multiple of 5 and 7, which is 35. The quotient of dividing 999 by 35 is 28, and the quotient of dividing 100 by 35 is 2. So, there are 28 - 2 + 1 = 27 numbers divisible by both 5 and 7.
To find the numbers divisible by both 5 and 9, we need to find the multiples of the least common multiple of 5 and 9, which is 45. The quotient of dividing 999 by 45 is 22, and the quotient of dividing 100 by 45 is 2. So, there are 22 - 2 + 1 = 21 numbers divisible by both 5 and 9.
To find the numbers divisible by both 7 and 9, we need to find the multiples of the least common multiple of 7 and 9, which is 63. The quotient of dividing 999 by 63 is 15, and the quotient of dividing 100 by 63 is 1. So, there are 15 - 1 + 1 = 15 numbers divisible by both 7 and 9.
To find the numbers divisible by 5, 7, and 9, we need to find the multiples of the least common multiple of 5, 7, and 9, which is 315. The quotient of dividing 999 by 315 is 3, and the quotient of dividing 100 by 315 is 0. So, there are 3 - 0 + 1 = 4 numbers divisible by 5, 7, and 9.
Finally, to find the total number of 3-digit numbers that are divisible either by 5, 7, or 9, we can add up the numbers we counted for each individual number and subtract the numbers that are counted more than once: 180 + 128 + 111 - 27 - 21 - 15 + 4 = 360