To solve the problem, we need to determine how many three-digit numbers are:

1. Not divisible by 45, and
2. Divisible by either 3 or 4.

Step 1: Count the total number of three-digit numbers.

The three-digit numbers range from 100 to 999. So, the total number of three-digit numbers is: $999 - 100 + 1 = 900$

Step 2: Count the number of three-digit numbers divisible by 45.

A number divisible by 45 must be divisible by both 5 and 9 (since $45 = 5 \times 9$). The least three-digit number divisible by 45 is 105, and the greatest is 990.

To find how many three-digit numbers are divisible by 45, we use: $\text{Number of terms} = \frac{\text{Last term} - \text{First term}}{\text{Common difference}} + 1$ $= \frac{990 - 105}{45} + 1 = \frac{885}{45} + 1 = 19 + 1 = 20$ So, there are 20 three-digit numbers divisible by 45.

Step 3: Count the number of three-digit numbers divisible by either 3 or 4.

• Divisible by 3: The smallest three-digit number divisible by 3 is 102, and the largest is 999. Using the same formula as above: $\text{Number of terms} = \frac{999 - 102}{3} + 1 = \frac{897}{3} + 1 = 299 + 1 = 300$
• Divisible by 4: The smallest three-digit number divisible by 4 is 100, and the largest is 996. Using the same formula: $\text{Number of terms} = \frac{996 - 100}{4} + 1 = \frac{896}{4} + 1 = 224 + 1 = 225$

Step 4: Use the principle of inclusion-exclusion to count the numbers divisible by either 3 or 4.

Some numbers are divisible by both 3 and 4 (i.e., divisible by 12). The smallest three-digit number divisible by 12 is 108, and the largest is 996. The count of such numbers is: $\text{Number of terms} = \frac{996 - 108}{12} + 1 = \frac{888}{12} + 1 = 74 + 1 = 75$ Now, the number of three-digit numbers divisible by either 3 or 4 is: $300 + 225 - 75 = 450$

Step 5: Subtract the numbers divisible by 45 from those divisible by either 3 or 4.

Numbers divisible by 45 are included in the 450 numbers divisible by either 3 or 4. We need to subtract these 20 numbers: $450 - 20 = 430$

So, there are 430 three-digit numbers that are not divisible by 45 but are divisible by either 3 or 4.

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Would you like more details on any step, or do you have further questions? Here are five related problems to explore:

1. How many three-digit numbers are divisible by 5 but not by 9?
2. What is the total number of three-digit numbers divisible by both 6 and 8?
3. How many three-digit numbers are not divisible by 3 but divisible by 7?
4. Find the number of three-digit numbers divisible by either 2, 3, or 5.
5. How many four-digit numbers are divisible by 45 but not by 90?

Tip: Understanding the principle of inclusion-exclusion can greatly simplify problems involving multiple conditions.