In the analogy “19 : 6859 :: 21 : ____”, which number should replace the blank to continue the same numerical pattern?

Introduction / Context:
This numerical analogy explores the relationship between a number and a much larger number derived from it. The pair “19 : 6859” suggests that 6859 is generated from 19 by some mathematical operation. To complete the analogy, you must determine that operation and apply it to 21, then select the correct result from the given options. This question mainly tests your familiarity with powers and cubes of integers.

Given Data / Assumptions:

• First numerical pair: 19 and 6859.
• Second starting number: 21 with an unknown corresponding value.
• Options: 9291, 9281, 9261, 9200, 9300.
• We assume standard integer powers and basic arithmetic.

Concept / Approach:
First, examine how 6859 is related to 19. A good strategy for such large numbers is to check squares and cubes of 19. Calculating 19^2 yields 361, which is much smaller than 6859. Calculating 19^3 gives 19 * 19 * 19, which equals 6859 exactly. So the pattern is “number → its cube”. To keep the analogy consistent, we must cube 21 to obtain the missing term. 21^3 = 21 * 21 * 21 = 9261. Hence, the correct answer is 9261.

Step-by-Step Solution:

Step 1: Compute the square of 19 to test if 6859 might be 19^2. 19^2 = 19 * 19 = 361, which is far smaller than 6859. Step 2: Compute the cube of 19. 19^3 = 19 * 19 * 19 = 6859, which matches the number in the analogy. Step 3: Conclude that the transformation is “take the cube of the given number.” Step 4: Apply the same rule to 21. First find 21^2 = 21 * 21 = 441. Step 5: Multiply again by 21 to obtain 21^3. 441 * 21 = 9261. Step 6: Match 9261 with the options and select it as the correct answer.

Verification / Alternative check:
Quickly inspect the other options. 9291, 9281, 9200, and 9300 are all close in value but do not match the exact cube of 21. Only 9261 equals 21^3. Since the first pair uses an exact cube, any other transformation (such as squaring or adding constants) would break the clean pattern already discovered.

Why Other Options Are Wrong:
The numbers 9291, 9281, 9200, and 9300 are not equal to any simple power of 21. They appear only as distractors around the correct cube value. Because the analogy depends on exactly cubing the given number, any value that does not match 21^3 is incorrect.

Common Pitfalls:
A common error is to approximate or guess based on the size of the numbers rather than calculating the cube precisely. Another is to assume a less common rule, such as 19 multiplied by some factor, without checking powers first. In numerical analogies where one number is much larger than the other, always test squares and cubes early. This strategy makes it much easier to identify the correct pattern and apply it correctly to the second pair.

Final Answer:
Since 6859 equals 19^3 and 21^3 equals 9261, the analogy “19 : 6859 :: 21 : ____” is completed as “19 : 6859 :: 21 : 9261”.