How many three-digit counting numbers

are exactly divisible by 6 but not also exactly divisible by 9?

all 3 digit number divisible by 6:

102 108 114 120 126 132 138 144...978 984 990 996
How many of those are there ?
(996-102)/6 + 1 = 150

or: An AG with a=102, d = 6, term(n) = 996
term(n) = a+(n-1)d
996 = 102 + 6(n-1)
6n-6 = 894
6n = 900
n = 150

Of those which ARE divisible by 9:
108 126 144 ... 990
This is another AG, where a = 108, d = 18, term(n) = 990
How many of those?
990 = 108 + 18(n-1)
882 = 18n - 18
18n = 900
n = 50
of those NOT divisible by 9 = 150 - 50 = 100

But why isn’t it 50????

900 / 6 = 150

900 / 9 = 100

150 - 100 = 50

To find the number of three-digit counting numbers that are divisible by 6 but not divisible by 9, we need to consider the divisibility rules of both numbers.

Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

Let's break down the problem step-by-step.

Step 1: Find the number of three-digit counting numbers divisible by 6.
We know that a number is divisible by 6 if it is divisible by both 2 and 3.
Since we are looking for three-digit numbers, the smallest three-digit number divisible by 6 is 102 (which is divisible by 2 and 3), and the largest three-digit number divisible by 6 is 996 (which is divisible by 2 and 3).
To find the count of these numbers, we can use the formula:

count = (last term - first term) / common difference + 1

First term = 102
Last term = 996
Common difference = 6 (since each number is increased by 6)

Plugging these values into the formula, we get:
count = (996 - 102) / 6 + 1
count = 894 / 6 + 1
count = 149 + 1
count = 150

So, there are 150 three-digit counting numbers that are divisible by 6.

Step 2: Find the number of three-digit counting numbers divisible by 9.
We know that a number is divisible by 9 if the sum of its digits is divisible by 9.
For a three-digit number to be divisible by 9, the sum of its digits must be a multiple of 9.
The smallest three-digit number whose digits add up to a multiple of 9 is 108, and the largest three-digit number whose digits add up to a multiple of 9 is 999.

To find the count of these numbers, we can use the formula discussed in Step 1:

count = (last term - first term) / common difference + 1

First term = 108
Last term = 999
Common difference = 9 (since each number is increased by 9)

Plugging these values into the formula, we get:
count = (999 - 108) / 9 + 1
count = 891 / 9 + 1
count = 99 + 1
count = 100

So, there are 100 three-digit counting numbers that are divisible by 9.

Step 3: Find the number of three-digit counting numbers divisible by both 6 and 9.
To find the number of three-digit counting numbers divisible by both 6 and 9, we need to find the common multiples of 6 and 9 within the three-digit range.

The smallest common multiple of 6 and 9 is 18, and the largest common multiple that is within the three-digit range is 990.

To find the count of these numbers, we can use the formula discussed in Step 1:

count = (last term - first term) / common difference + 1

First term = 18
Last term = 990
Common difference = 18 (since each number is increased by 18)

Plugging these values into the formula, we get:
count = (990 - 18) / 18 + 1
count = 972 / 18 + 1
count = 54 + 1
count = 55

So, there are 55 three-digit counting numbers that are divisible by both 6 and 9.

Step 4: Find the number of three-digit counting numbers divisible by 6 but not divisible by 9.
To find the number of three-digit counting numbers that are divisible by 6 but not divisible by 9, we need to subtract the number of three-digit counting numbers divisible by both 6 and 9 (Step 3) from the total number of three-digit counting numbers divisible by 6 (Step 1).

Number of three-digit counting numbers divisible by 6 but not divisible by 9 = Total number of three-digit counting numbers divisible by 6 - Number of three-digit counting numbers divisible by both 6 and 9

= 150 - 55
= 95

Therefore, there are 95 three-digit counting numbers that are exactly divisible by 6 but not divisible by 9.

To find the number of three-digit counting numbers that are divisible by 6 but not by 9, we need to first determine the counting numbers between 100 and 999 that are divisible by 6, and then exclude those that are also divisible by 9.

Step 1: Finding the counting numbers divisible by 6
To find the counting numbers divisible by 6, we need to identify the first and last three-digit numbers divisible by 6.

The first three-digit number divisible by 6 is 102 (6 x 17 = 102)
The last three-digit number divisible by 6 is 996 (6 x 166 = 996)

So the counting numbers divisible by 6 between 100 and 999 are 102, 108, ..., 996.

Step 2: Excluding numbers divisible by 9
Next, we need to identify the counting numbers from the list obtained in Step 1 that are divisible by 9. We can use a simple divisibility rule: a number is divisible by 9 if the sum of its digits is divisible by 9.

To find the sum of the digits, we can add the individual digits of each number. For example, the sum of the digits in 102 is 1 + 0 + 2 = 3.

If the sum of the digits of a number from the list obtained in Step 1 is divisible by 9, we exclude that number.

Step 3: Counting the remaining numbers
Count the numbers that are left after excluding the digits divisible by 9.

So, go through the list obtained in Step 1 and exclude the numbers that are divisible by 9 (using the sum of digits rule). Now count the remaining numbers, and you will have the answer to the question.

Note: You can perform these calculations manually or use programming tools like Python to automate the process.