How many numbers are there from 100 to 285 which are divisible by 5?

Options

A. 40
B. 38
C. 35
D. 42

Answer

To find how many numbers are there from 100 to 285 that are divisible by 5, we can follow these steps:

### Step 1: Identify the first and last terms
The first number in the range that is divisible by 5 is 100, and the last number in the range that is divisible by 5 is 285.

### Step 2: Establish the common difference
Since we are looking for numbers divisible by 5, the common difference (d) is 5. This means the sequence of numbers divisible by 5 will be: 100, 105, 110, 115, ..., 285.

### Step 3: Use the formula for the nth term of an arithmetic progression (AP)
The nth term of an AP can be calculated using the formula:
\[ a_n = a + (n - 1) \cdot d \]
Where:
- \( a_n \) is the nth term,
- \( a \) is the first term,
- \( n \) is the number of terms,
- \( d \) is the common difference.

In our case:
- \( a_n = 285 \) (the last term),
- \( a = 100 \) (the first term),
- \( d = 5 \) (the common difference).

### Step 4: Set up the equation
Substituting the known values into the formula gives:
\[ 285 = 100 + (n - 1) \cdot 5 \]

### Step 5: Solve for n
First, simplify the equation:
\[ 285 - 100 = (n - 1) \cdot 5 \]
\[ 185 = (n - 1) \cdot 5 \]

Now, divide both sides by 5:
\[ n - 1 = \frac{185}{5} \]
\[ n - 1 = 37 \]

Now, add 1 to both sides to find n:
\[ n = 37 + 1 \]
\[ n = 38 \]

### Conclusion
Therefore, there are a total of **38 numbers** from 100 to 285 that are divisible by 5.

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