How many numbers are there from 209 to 285 divisible by both 9 and 4?

Options

A. 2
B. 3
C. 4
D. 5

Answer

To find how many numbers between 209 and 285 are divisible by both 9 and 4, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
To determine the numbers that are divisible by both 9 and 4, we first need to find the LCM of these two numbers.

- The prime factorization of 9 is \(3^2\).
- The prime factorization of 4 is \(2^2\).

The LCM is found by taking the highest power of each prime factor:
- For 3, the highest power is \(3^2\).
- For 2, the highest power is \(2^2\).

Thus, the LCM of 9 and 4 is:
\[ LCM(9, 4) = 3^2 \times 2^2 = 9 \times 4 = 36 \]

### Step 2: Identify the Range
We need to find numbers divisible by 36 between 209 and 285.

### Step 3: Find the First Multiple of 36 Greater than or Equal to 209
To find the first multiple of 36 that is greater than or equal to 209, we can divide 209 by 36 and round up to the nearest whole number:
\[ \frac{209}{36} \approx 5.8056 \]
Rounding up gives us 6. Now, we multiply 6 by 36:
\[ 6 \times 36 = 216 \]

### Step 4: Find the Last Multiple of 36 Less than or Equal to 285
Next, we find the largest multiple of 36 that is less than or equal to 285. We divide 285 by 36:
\[ \frac{285}{36} \approx 7.9167 \]
Rounding down gives us 7. Now, we multiply 7 by 36:
\[ 7 \times 36 = 252 \]

### Step 5: List the Multiples of 36 Between 209 and 285
Now we have the first multiple (216) and the last multiple (252). The multiples of 36 in this range are:
- 216
- 252

### Step 6: Count the Numbers
The multiples of 36 between 209 and 285 are 216 and 252. Thus, there are 2 numbers.

### Final Answer
The total count of numbers between 209 and 285 that are divisible by both 9 and 4 is **2**.

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