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

Options

A. a) 16
B. b) 18
C. c) 19
D. 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 numbers between 1000 and 5000.

### Step 2: Find the smallest number greater than or equal to 1000 that is divisible by 225
To find the smallest number greater than or equal to 1000 that is divisible by 225, we can use the formula:
\[ \text{Smallest number} = \lceil \frac{1000}{225} \rceil \times 225 \]

Calculating:
\[ \frac{1000}{225} \approx 4.44 \]
Taking the ceiling gives us 5.
So, the smallest number is:
\[ 5 \times 225 = 1125 \]

### Step 3: Find the largest number less than or equal to 5000 that is divisible by 225
To find the largest number less than or equal to 5000 that is divisible by 225, we can use the formula:
\[ \text{Largest number} = \lfloor \frac{5000}{225} \rfloor \times 225 \]

Calculating:
\[ \frac{5000}{225} \approx 22.22 \]
Taking the floor gives us 22.
So, the largest number is:
\[ 22 \times 225 = 4950 \]

### Step 4: List the numbers divisible by 225 between 1125 and 4950
The numbers divisible by 225 in this range form an arithmetic sequence where:
- First term (a) = 1125
- Common difference (d) = 225
- Last term (l) = 4950

### Step 5: Find the number of terms in the sequence
The number of terms (n) in an arithmetic sequence can be found using the formula:
\[ n = \frac{l - a}{d} + 1 \]

Substituting the values:
\[ n = \frac{4950 - 1125}{225} + 1 \]
\[ n = \frac{3825}{225} + 1 \]
\[ n = 17 + 1 \]
\[ n = 18 \]

### Conclusion
There are **18 numbers** between 1000 and 5000 that are exactly divisible by 225.

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