How many three-digit numbers are divisible by 7?

To find how many three-digit numbers are divisible by 7, we first need to identify the range of three-digit numbers and then determine which of these are divisible by 7.

Identifying the Range of Three-Digit Numbers

Three-digit numbers range from 100 to 999. Our goal is to find all numbers within this range that can be divided evenly by 7.

Finding the First Three-Digit Number Divisible by 7

We start by identifying the smallest three-digit number that is divisible by 7. To do this, we can divide 100 by 7 and round up to the nearest whole number:

• 100 ÷ 7 = 14.2857 (approximately)
• Rounding up gives us 15.

Now, we multiply 15 by 7 to find the first three-digit number:

• 15 × 7 = 105.

Thus, the first three-digit number divisible by 7 is 105.

Finding the Last Three-Digit Number Divisible by 7

Next, we need to find the largest three-digit number that is divisible by 7. We can do this by dividing 999 by 7 and rounding down:

• 999 ÷ 7 = 142.7143 (approximately)
• Rounding down gives us 142.

Now, we multiply 142 by 7 to find the last three-digit number:

• 142 × 7 = 994.

Therefore, the last three-digit number divisible by 7 is 994.

Calculating the Total Count of Three-Digit Numbers Divisible by 7

Now that we have the first three-digit number (105) and the last three-digit number (994), we can find how many numbers lie between them that are divisible by 7. The numbers divisible by 7 form an arithmetic sequence where:

• The first term (a) is 105.
• The last term (l) is 994.
• The common difference (d) is 7.

Using the Formula for the Number of Terms in an Arithmetic Sequence

The formula to find the number of terms (n) in an arithmetic sequence is:

n = (l - a) / d + 1

Substituting our values into the formula:

• n = (994 - 105) / 7 + 1
• n = 889 / 7 + 1
• n = 127 + 1
• n = 128.

Final Result

In conclusion, there are 128 three-digit numbers that are divisible by 7. This method of breaking down the problem into manageable steps helps ensure accuracy and clarity in finding the solution.