How many three-digit numbers are divisible by 7?
128
Step 1: Identify the range of three-digit numbers.
The smallest three-digit number is 100 and the largest is 999.
Step 2: Find the smallest three-digit number divisible by 7.
Divide 100 by 7: \(100 \div 7 \approx 14.28\). The next whole number is 15.
So the first three-digit multiple of 7 is:
\[7 \times 15 = 105\]
Step 3: Find the largest three-digit number divisible by 7.
Divide 999 by 7: \(999 \div 7 \approx 142.71\). Take the whole number part 142.
So the last three-digit multiple of 7 is:
\[7 \times 142 = 994\]
Step 4: Observe that these form an AP.
The numbers divisible by 7 between 100 and 999 form an arithmetic progression:
\[105, 112, 119, \ldots, 994\]
Here, first term \(a = 105\), common difference \(d = 7\), last term \(l = 994\).
Step 5: Use the formula for the number of terms in an AP.
For an AP, the number of terms \(n\) is:
\[n = \frac{l - a}{d} + 1\]
Substitute the values:
\[n = \frac{994 - 105}{7} + 1\]
\[n = \frac{889}{7} + 1\]
\[n = 127 + 1 = 128\]
Conclusion: There are 128 three-digit numbers that are divisible by 7.