To solve the problem of how many numbers between 500 and 900 (inclusive) are exactly divisible by 12, 15, 20, and 30, we can follow these steps:
### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the LCM of the numbers 12, 15, 20, and 30. The LCM is the smallest number that is divisible by all the given numbers.
- **Prime factorization**:
- 12 = 2^2 * 3^1
- 15 = 3^1 * 5^1
- 20 = 2^2 * 5^1
- 30 = 2^1 * 3^1 * 5^1
- **Take the highest power of each prime factor**:
- For 2: the highest power is 2^2 (from 12 and 20)
- For 3: the highest power is 3^1 (from 12, 15, and 30)
- For 5: the highest power is 5^1 (from 15, 20, and 30)
- **Calculate the LCM**:
\[
\text{LCM} = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60
\]
### Step 2: Identify the Range
Next, we need to find the multiples of 60 that lie between 500 and 900, inclusive.
### Step 3: Find the First Multiple of 60 Greater Than or Equal to 500
To find the smallest multiple of 60 that is greater than or equal to 500:
\[
\text{First multiple} = 60 \times \lceil \frac{500}{60} \rceil
\]
Calculating:
\[
\frac{500}{60} \approx 8.33 \Rightarrow \lceil 8.33 \rceil = 9
\]
So, the first multiple is:
\[
60 \times 9 = 540
\]
### Step 4: Find the Last Multiple of 60 Less Than or Equal to 900
To find the largest multiple of 60 that is less than or equal to 900:
\[
\text{Last multiple} = 60 \times \lfloor \frac{900}{60} \rfloor
\]
Calculating:
\[
\frac{900}{60} = 15 \Rightarrow \lfloor 15 \rfloor = 15
\]
So, the last multiple is:
\[
60 \times 15 = 900
\]
### Step 5: Count the Multiples of 60 Between 540 and 900
Now, we need to count the multiples of 60 from 540 to 900:
- The multiples of 60 in this range are:
- 540 (60 x 9)
- 600 (60 x 10)
- 660 (60 x 11)
- 720 (60 x 12)
- 780 (60 x 13)
- 840 (60 x 14)
- 900 (60 x 15)
### Step 6: Total Count
Counting these multiples gives us:
- 540, 600, 660, 720, 780, 840, 900 (7 numbers)
### Final Answer
Thus, the total number of numbers between 500 and 900 that are exactly divisible by 12, 15, 20, and 30 is **7**.
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