How many 3-digit numbers are completely divided by 4.

Options

A. 225
B. 180
C. 223
D. 178

Answer

To find how many 3-digit numbers are completely divisible by 4, 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: Find the smallest 3-digit number divisible by 4
To find the smallest 3-digit number divisible by 4, we can divide 100 by 4:
\[ 100 \div 4 = 25 \]
Since 100 is divisible by 4, the smallest 3-digit number divisible by 4 is **100**.

### Step 3: Find the largest 3-digit number divisible by 4
To find the largest 3-digit number divisible by 4, we can divide 999 by 4:
\[ 999 \div 4 = 249.75 \]
Taking the floor of 249.75 gives us 249. Now, we multiply back by 4 to find the largest 3-digit number divisible by 4:
\[ 249 \times 4 = 996 \]
So, the largest 3-digit number divisible by 4 is **996**.

### Step 4: Determine the sequence of 3-digit numbers divisible by 4
The sequence of 3-digit numbers divisible by 4 starts from 100 and ends at 996. The common difference \(D\) is 4 (since we are looking at numbers divisible by 4).

### Step 5: Use the formula for the nth term of an arithmetic progression (AP)
The nth term of an AP can be expressed as:
\[ a_n = a + (n-1) \cdot d \]
Where:
- \(a\) is the first term (100),
- \(d\) is the common difference (4),
- \(a_n\) is the last term (996).

Setting up the equation:
\[ 996 = 100 + (n-1) \cdot 4 \]

### Step 6: Solve for \(n\)
Rearranging the equation:
\[ 996 - 100 = (n-1) \cdot 4 \]
\[ 896 = (n-1) \cdot 4 \]
Now, divide both sides by 4:
\[ n-1 = 224 \]
Adding 1 to both sides gives:
\[ n = 225 \]

### Conclusion
Thus, the total number of 3-digit numbers that are completely divisible by 4 is **225**.

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