Which of the following statement(s) is/are true?
I. There are 12 multiples of 9 from 7 to 109.
II. There are 9 multiples of 13 from 19 to 119.

Options

A. Only I
B. Only II
C. Both I and II
D. Neither I nor II

Answer

To determine the truth of the statements regarding the multiples of 9 and 13, we will analyze each statement step by step.

### Step 1: Analyze Statement I
**Statement I:** There are 12 multiples of 9 from 7 to 109.

1. **Identify the range:** We need to find the multiples of 9 that lie between 7 and 109.
2. **Find the first multiple of 9 greater than or equal to 7:** The first multiple of 9 is 9 itself.
3. **Find the last multiple of 9 less than or equal to 109:** To find this, we can divide 109 by 9 and take the floor of the result.
\[ \frac{109}{9} \approx 12.11 \implies \text{The largest integer is } 12. \]
Therefore, the last multiple is:
\[ 9 \times 12 = 108. \]
4. **List the multiples of 9 from 9 to 108:**
- The multiples are: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108.
5. **Count the multiples:** There are 12 multiples of 9.

**Conclusion for Statement I:** True.

### Step 2: Analyze Statement II
**Statement II:** There are 9 multiples of 13 from 19 to 119.

1. **Identify the range:** We need to find the multiples of 13 that lie between 19 and 119.
2. **Find the first multiple of 13 greater than or equal to 19:** The first multiple of 13 is:
\[ 13 \times 2 = 26. \]
3. **Find the last multiple of 13 less than or equal to 119:** To find this, we can divide 119 by 13 and take the floor of the result.
\[ \frac{119}{13} \approx 9.15 \implies \text{The largest integer is } 9. \]
Therefore, the last multiple is:
\[ 13 \times 9 = 117. \]
4. **List the multiples of 13 from 26 to 117:**
- The multiples are: 26, 39, 52, 65, 78, 91, 104, 117.
5. **Count the multiples:** There are 8 multiples of 13.

**Conclusion for Statement II:** False.

### Final Conclusion
- **Statement I is true.**
- **Statement II is false.**

### Summary
Only Statement I is true.