how many different counting numbers less than 200 are exactly divisible by either 6 or 9 or both?

Answer 1

According to a source, there are 44 counting numbers less than 200 that are exactly divisible by either 6 or 9, or by both.

To determine the total count, we can follow these steps:

Find out how many counting numbers less than 200 are divisible by 6. The last number under 200 that is divisible by 6 is 198, and since 198 is the 33rd multiple of 6, there are 33 such numbers.

Next, figure out how many numbers are divisible by 9. The last number under 200 that is divisible by 9 is also 198, and since 198 is the 22nd multiple of 9, there are 22 such numbers.

Some numbers will be divisible by both 6 and 9, but we need to avoid counting these twice. So, determine which numbers are divisible by both (these are actually multiples of 18). The last number under 200 that is divisible by 18 is also 198, and since it is the 11th multiple of 18, there are 11 such numbers.

Finally, add the two individual counts from steps 1 and 2 together and subtract the count from step 3 to eliminate double counting: 33 + 22 - 11 = 44.

Therefore, there are 44 different counting numbers less than 200 that are exactly divisible by either 6 or 9 or both.