In the number analogy "3 : 243 :: 5 : ____", which number correctly completes the pattern using an exponent rule?

Introduction / Context:
This analogy is based on powers of numbers. The pair "3 : 243" suggests that 243 is generated from 3 by raising it to a certain power. You must identify this exponent and then apply the same exponent to 5 to find the missing number. Such problems test your knowledge of standard powers like squares, cubes, and higher powers, especially for small integers.

Given Data / Assumptions:

• First pair: 3 corresponds to 243.
• Second pair: 5 corresponds to an unknown number.
• Options: 125, 625, 3025, 3125, 243.
• We look for a simple power relationship such as n^4 or n^5.

Concept / Approach:
We recall that 3^3 = 27 and 3^4 = 81. Neither of these is equal to 243. Calculate 3^5: 3 * 3 * 3 * 3 * 3 = 243. Thus 243 is exactly 3 raised to the power of 5. The rule seems to be "second number equals first number raised to the power 5." To apply this rule to 5, we compute 5^5. Knowing that 5^2 = 25 and 5^3 = 125, we can compute higher powers from these.

Step-by-Step Solution:

Step 1: Compute 3^3 = 27 and 3^4 = 81, observe that neither equals 243. Step 2: Compute 3^5 = 3 * 3 * 3 * 3 * 3 = 243, which matches the given second number. Step 3: Conclude that the mapping is "n maps to n^5". Step 4: Compute 5^2 = 25 and 5^3 = 125. Step 5: Compute 5^4 = 5 * 125 = 625. Step 6: Compute 5^5 = 5 * 625 = 3125. Step 7: Select 3125 from the options as the correct answer.

Verification / Alternative check:
Check the other options. 125 is 5^3, and 625 is 5^4, but the relationship given in the first pair involves the 5th power, not the 3rd or 4th. 3025 does not equal any simple power like 5^4 or 5^5. 243 is 3^5, already used in the first pair, and is not related to 5. Only 3125 matches exactly as 5^5, keeping the pattern consistent for both pairs.

Why Other Options Are Wrong:
Choosing 125 or 625 would implicitly change the rule to "cube" or "fourth power", but those rules do not produce 243 from 3. 3025 and 243 fail to correspond to any consistent power mapping from 5 that matches the mapping from 3 to 243. Since analogies require the same operation for both pairs, these alternatives must be discarded.

Common Pitfalls:
Many test takers stop at 5^3 = 125 or 5^4 = 625 and pick these values because they are more familiar. They forget to confirm which power actually produced the first pair. Another pitfall is to assume that the power must be 3, because cubes are common, and ignore 3^5. Careful checking of the first pair prevents such mistakes.

Final Answer:
Since 243 equals 3^5 and the same exponent must be used, the corresponding number for 5 is 5^5 = 3125, completing the analogy "3 : 243 :: 5 : 3125".