# Determining the Number of Three-Digit Numbers Divisible by 7

## There are 128 three-digit numbers that are divisible by 7. This comprehensive approach ensures you accurately determine the count of such numbers

To find out how many three-digit numbers are divisible by 7, follow these systematic steps:

### Step 1: Identify the Smallest and Largest Three-Digit Numbers Divisible by 7

**Finding the Smallest Three-Digit Number Divisible by 7:**

- Start with the smallest three-digit number, which is 100.
- To determine the smallest three-digit number divisible by 7, divide 100 by 7: $\frac{100}{7} \approx 14.2857$ Round up to the next whole number, which is 15.
- Multiply 15 by 7 to find the smallest three-digit number: $15 \times 7 = 105$ Therefore, 105 is the smallest three-digit number divisible by 7.

**Finding the Largest Three-Digit Number Divisible by 7:**

- Start with the largest three-digit number, which is 999.
- To determine the largest three-digit number divisible by 7, divide 999 by 7: $\frac{999}{7} \approx 142.7143$ Round down to the nearest whole number, which is 142.
- Multiply 142 by 7 to find the largest three-digit number: $142 \times 7 = 994$ Therefore, 994 is the largest three-digit number divisible by 7.

### Step 2: Formulate the Arithmetic Sequence

The sequence of three-digit numbers divisible by 7 is an arithmetic sequence with:

- The first term $a = 105$
- The common difference $d = 7$

### Step 3: Calculate the Number of Terms in the Sequence

To determine the number of terms, we use the formula for the $n$-th term of an arithmetic sequence:

$a_n = a + (n-1)d$

Where:

- $a_n$ is the last term (994 in this case)
- $a$ is the first term (105)
- $d$ is the common difference (7)

Substitute the values into the formula and solve for $n$:

$994 = 105 + (n-1) \times 7$

Subtract 105 from both sides:

$889 = (n-1) \times 7$

Divide both sides by 7:

$127 = n-1$

Solve for $n$:

$n = 127 + 1$ $n = 128$

### Conclusion

There are **128** three-digit numbers that are divisible by 7. This comprehensive approach ensures you accurately determine the count of such numbers, demonstrating the power of arithmetic sequences in solving real-world numerical problems.