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 :
5:9 :: 10:?

Options

A. 17
B. 28
C. 65
D. 126

Explanation

To solve the analogy problem 5:9 :: 10:?, we need to identify the relationship between the numbers in the first pair (5 and 9) and apply the same relationship to the second pair (10 and ?).

### Step-by-Step Solution:

1. **Identify the relationship between 5 and 9:**
- We notice that 5 can be represented as \(2^2 + 1\).
- Calculation: \(2^2 = 4\) and \(4 + 1 = 5\).
- Next, for 9, we can represent it as \(2^3 + 1\).
- Calculation: \(2^3 = 8\) and \(8 + 1 = 9\).

2. **Establish the pattern:**
- The first number (5) is derived from squaring 2 and adding 1.
- The second number (9) is derived from cubing 2 and adding 1.

3. **Apply the same pattern to the number 10:**
- We need to find a number that relates to 10 in the same way.
- To find the first number (10), we can represent it as \(3^2 + 1\).
- Calculation: \(3^2 = 9\) and \(9 + 1 = 10\).

4. **Now, apply the same pattern to find the second number:**
- We need to find the number that corresponds to 10 using the cube of 3.
- Calculation: \(3^3 + 1\).
- First, calculate \(3^3 = 27\).
- Then, add 1: \(27 + 1 = 28\).

5. **Conclusion:**
- Therefore, the number that corresponds to 10 in the same way that 9 corresponds to 5 is 28.

### Final Answer:
The answer is 28.