How many three digit numbers are divisible by 6?

Options

A. 196
B. 149
C. 150
D. 151

Answer

To find how many three-digit numbers are divisible by 6, we need to follow these steps:

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

### Step 2: Determine the smallest three-digit number divisible by 6
To find the smallest three-digit number that is divisible by 6, we can divide 100 by 6 and round up to the nearest whole number, then multiply by 6.

\[ 100 \div 6 = 16.67 \quad \text{(round up to 17)} \]
\[ 17 \times 6 = 102 \]

So, the smallest three-digit number divisible by 6 is **102**.

### Step 3: Determine the largest three-digit number divisible by 6
To find the largest three-digit number that is divisible by 6, we can divide 999 by 6 and round down to the nearest whole number, then multiply by 6.

\[ 999 \div 6 = 166.5 \quad \text{(round down to 166)} \]
\[ 166 \times 6 = 996 \]

So, the largest three-digit number divisible by 6 is **996**.

### Step 4: Find the total count of three-digit numbers divisible by 6
Now we need to find how many numbers are there from 102 to 996 that are divisible by 6. This forms an arithmetic sequence where:
- The first term \(a = 102\)
- The last term \(l = 996\)
- The common difference \(d = 6\)

The formula for the \(n\)-th term of an arithmetic sequence is given by:
\[ l = a + (n - 1) \cdot d \]
Rearranging this to find \(n\):
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values we have:
\[ n = \frac{996 - 102}{6} + 1 \]
\[ n = \frac{894}{6} + 1 \]
\[ n = 149 + 1 = 150 \]

Thus, the total number of three-digit numbers that are divisible by 6 is **150**.

### Summary of Steps:
1. Identify the range of three-digit numbers (100 to 999).
2. Find the smallest three-digit number divisible by 6 (102).
3. Find the largest three-digit number divisible by 6 (996).
4. Calculate the total count of three-digit numbers divisible by 6 (150).