How many 3-digit numbers are divisible by 5 ?

Options

A. (a) 160
B. (b) 165
C. (c) 175
D. (d) 180

Answer

To find how many 3-digit numbers are divisible by 5, we can follow these steps:

### Step 1: Identify the range of 3-digit numbers
The smallest 3-digit number is 100 and the largest is 999.

### Step 2: Determine the first 3-digit number divisible by 5
The first 3-digit number is 100. Since 100 is divisible by 5 (100 ÷ 5 = 20), we can take 100 as our starting point.

### Step 3: Determine the last 3-digit number divisible by 5
The largest 3-digit number is 999. To find the largest 3-digit number divisible by 5, we can check the largest number less than or equal to 999 that is divisible by 5.
999 ÷ 5 = 199.8, so the largest integer is 199. Multiplying back, 199 × 5 = 995. Thus, the last 3-digit number divisible by 5 is 995.

### Step 4: Identify the sequence of numbers
Now we have a sequence of numbers starting from 100 to 995, where each number is divisible by 5. This forms an arithmetic progression (AP) where:
- First term (A) = 100
- Common difference (D) = 5
- Last term (L) = 995

### Step 5: Find the number of terms (N) in the sequence
The formula for the nth term of an arithmetic progression is given by:
\[ L = A + (N - 1)D \]
Substituting the known values:
\[ 995 = 100 + (N - 1) \cdot 5 \]

### Step 6: Solve for N
1. Rearranging the equation:
\[ 995 - 100 = (N - 1) \cdot 5 \]
\[ 895 = (N - 1) \cdot 5 \]

2. Dividing both sides by 5:
\[ N - 1 = 179 \]

3. Adding 1 to both sides:
\[ N = 180 \]

### Conclusion
Thus, there are **180** three-digit numbers that are divisible by 5.

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