We are tasked with finding how many integers between 100 and 500 have the numbers 6, 9, 10, and 45 as factors.

Step 1: Find the Least Common Multiple (LCM)

To solve this, we need to determine the Least Common Multiple (LCM) of the given numbers, because any number that is divisible by all four numbers must also be divisible by their LCM.

The numbers are:

• 6 = $2 \times 3$
• 9 = $3^2$
• 10 = $2 \times 5$
• 45 = $3^2 \times 5$

To compute the LCM, we take the highest powers of all prime factors:

• For 2, the highest power is $2^1$ (from 6 and 10).
• For 3, the highest power is $3^2$ (from 9 and 45).
• For 5, the highest power is $5^1$ (from 10 and 45).

Thus, the LCM is: $\text{LCM}(6, 9, 10, 45) = 2^1 \times 3^2 \times 5^1 = 2 \times 9 \times 5 = 90$

Step 2: Find Multiples of 90 Between 100 and 500

Next, we need to find how many integers between 100 and 500 are divisible by 90.

• The smallest multiple of 90 greater than or equal to 100 is: $\lceil \frac{100}{90} \rceil = 2 \quad \Rightarrow \quad 90 \times 2 = 180$

• The largest multiple of 90 less than or equal to 500 is: $\lfloor \frac{500}{90} \rfloor = 5 \quad \Rightarrow \quad 90 \times 5 = 450$

Now, the multiples of 90 between 100 and 500 are: $180, 270, 360, 450$

Step 3: Count the Multiples

There are 4 multiples of 90 between 100 and 500.

Final Answer:

There are 4 integers between 100 and 500 that have 6, 9, 10, and 45 as factors.

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Would you like more details on any part of this solution? Here are 5 related questions to consider:

1. How do we calculate the LCM for more complex numbers?
2. What are the differences between LCM and GCD (Greatest Common Divisor)?
3. How would the answer change if the range was extended beyond 500?
4. Can prime factorization be used to solve divisibility problems in other contexts?
5. How would you find how many numbers in a range are divisible by a set of numbers?

Tip: When dealing with divisibility problems, always consider using the LCM for multiple factors.