Determining the Number of Three-Digit Numbers Divisible by 7

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:

1. Start with the smallest three-digit number, which is 100.
2. 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.
3. 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:

1. Start with the largest three-digit number, which is 999.
2. 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.
3. 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.