Based on the analogy given, find the missing pair form the given options
32 : 13 : : ___________ : __________
A. 51 : 36
B. 83 : 121
C. 71 : 81
D. 47 : 65

Options

A. D
B. C
C. B
D. A

Answer

To solve the analogy problem \(32 : 13 :: \_\_\_\_\_\_ : \_\_\_\_\_\_\), we need to find a relationship between the numbers in the first pair and apply the same relationship to the options provided.

### Step-by-Step Solution:

1. **Identify the relationship in the first pair (32 : 13)**:
- Split 32 into its digits: 3 and 2.
- Calculate \(3^2 + 2^2\):
\[ 3^2 = 9 \quad \text{and} \quad 2^2 = 4 \]
\[ 9 + 4 = 13 \]
- So, the relationship is that the second number (13) is the sum of the squares of the digits of the first number (32).

2. **Apply the same relationship to the options**:
- **Option A: 51 : 36**
- Split 51 into its digits: 5 and 1.
- Calculate \(5^2 + 1^2\):
\[ 5^2 = 25 \quad \text{and} \quad 1^2 = 1 \]
\[ 25 + 1 = 26 \quad (\text{not equal to } 36) \]
- **Option B: 83 : 121**
- Split 83 into its digits: 8 and 3.
- Calculate \(8^2 + 3^2\):
\[ 8^2 = 64 \quad \text{and} \quad 3^2 = 9 \]
\[ 64 + 9 = 73 \quad (\text{not equal to } 121) \]
- **Option C: 71 : 81**
- Split 71 into its digits: 7 and 1.
- Calculate \(7^2 + 1^2\):
\[ 7^2 = 49 \quad \text{and} \quad 1^2 = 1 \]
\[ 49 + 1 = 50 \quad (\text{not equal to } 81) \]
- **Option D: 47 : 65**
- Split 47 into its digits: 4 and 7.
- Calculate \(4^2 + 7^2\):
\[ 4^2 = 16 \quad \text{and} \quad 7^2 = 49 \]
\[ 16 + 49 = 65 \quad (\text{equal to } 65) \]

3. **Conclusion**:
- The correct option that follows the same relationship as \(32 : 13\) is **Option D: 47 : 65**.