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
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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: Round up to the next whole number, which is 15.
- Multiply 15 by 7 to find the smallest three-digit number: 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: Round down to the nearest whole number, which is 142.
- Multiply 142 by 7 to find the largest three-digit number: 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
- The common difference
Step 3: Calculate the Number of Terms in the Sequence
To determine the number of terms, we use the formula for the -th term of an arithmetic sequence:
Where:
- is the last term (994 in this case)
- is the first term (105)
- is the common difference (7)
Substitute the values into the formula and solve for :
Subtract 105 from both sides:
Divide both sides by 7:
Solve for :
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.