In the questions, there is a certain relationship between the two given numbers on one side of :: and one number is given on another side of :: . It is required to find another number from the given alternative having the same relationship with this number as the numbers of the given pair has :
10:65: 17:?

Options

A. 126
B. 82
C. 69
D. 111

Explanation

To solve the analogy problem given, we need to identify the relationship between the numbers 10 and 65, and then apply that same relationship to the number 17 to find the missing number.

### Step-by-Step Solution:

1. **Identify the relationship between 10 and 65**:
- The first number is 10. We can express 10 as \(3^2 + 1\):
\[ 3^2 = 9 \quad \text{and} \quad 9 + 1 = 10 \]
- The second number is 65. We can express 65 as \(4^3 + 1\):
\[ 4^3 = 64 \quad \text{and} \quad 64 + 1 = 65 \]

2. **Determine the base numbers used**:
- From the calculations, we see that the base numbers used are 3 for the first number (10) and 4 for the second number (65).

3. **Apply the same logic to the number 17**:
- The number 17 can be expressed as \(4^2 + 1\):
\[ 4^2 = 16 \quad \text{and} \quad 16 + 1 = 17 \]

4. **Find the corresponding number for 17**:
- We will use the next integer, which is 5, to find the missing number:
\[ 5^3 + 1 \]
- Calculate \(5^3\):
\[ 5^3 = 125 \]
- Now add 1:
\[ 125 + 1 = 126 \]

5. **Conclusion**:
- Therefore, the number that corresponds to 17, following the same relationship as between 10 and 65, is **126**.

### Final Answer:
The answer is **126**.