How many numbers between 1000 and 5000 are exactly divisible by 225 ?

Options

A. 16
B. 18
C. 19
D. 12

Answer

To find how many numbers between 1000 and 5000 are exactly divisible by 225, we can follow these steps:

### Step 1: Identify the range
We need to find all the numbers between 1000 and 5000 that are divisible by 225.

### Step 2: Find the smallest number greater than or equal to 1000 that is divisible by 225
To find this, we divide 1000 by 225 and round up to the nearest whole number:
\[ \text{Quotient} = \frac{1000}{225} \approx 4.44 \]
Rounding up gives us 5. Now we multiply 5 by 225 to find the smallest number:
\[ 5 \times 225 = 1125 \]

### Step 3: Find the largest number less than or equal to 5000 that is divisible by 225
Next, we divide 5000 by 225 and round down to the nearest whole number:
\[ \text{Quotient} = \frac{5000}{225} \approx 22.22 \]
Rounding down gives us 22. Now we multiply 22 by 225 to find the largest number:
\[ 22 \times 225 = 4950 \]

### Step 4: Count the numbers divisible by 225 between 1125 and 4950
The numbers divisible by 225 between 1125 and 4950 can be expressed as:
\[ 225 \times 5, 225 \times 6, 225 \times 7, \ldots, 225 \times 22 \]
This is an arithmetic sequence where:
- The first term \(a = 5\)
- The last term \(l = 22\)

### Step 5: Calculate the total count of terms
To find the total number of terms in this sequence, we use the formula for the number of terms in an arithmetic sequence:
\[ \text{Number of terms} = l - a + 1 \]
Substituting the values:
\[ \text{Number of terms} = 22 - 5 + 1 = 18 \]

### Conclusion
Thus, the total number of numbers between 1000 and 5000 that are exactly divisible by 225 is **18**.

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