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 :
3 : 243 :: 5 :?

Options

A. 425
B. 465
C. 546
D. 3125

Explanation

To solve the analogy problem given as \(3 : 243 :: 5 : ?\), we need to identify the relationship between the numbers in the first pair and apply the same relationship to the second pair.

### Step-by-Step Solution:

1. **Identify the relationship between 3 and 243**:
- We can express 243 as a power of 3.
- Calculate \(3^5\):
\[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \]
- Thus, the relationship is that the second number (243) is equal to the first number (3) raised to the power of 5.

2. **Apply the same relationship to the number 5**:
- Now, we need to find the number that corresponds to 5 using the same power of 5.
- Calculate \(5^5\):
\[ 5^5 = 5 \times 5 \times 5 \times 5 \times 5 \]
- We can break it down:
\[ 5 \times 5 = 25 \]
\[ 25 \times 5 = 125 \]
\[ 125 \times 5 = 625 \]
\[ 625 \times 5 = 3125 \]

3. **Conclusion**:
- Therefore, the number that corresponds to 5, following the same relationship as in the first pair, is \(3125\).

### Final Answer:
The answer is \(3125\).

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