Select the related number: 841 is to 29 as 289 is to which of the following numbers?

Introduction / Context:
This question is a number analogy based on squares and square roots. The pair 841 : 29 is provided, and you are asked to find the number that stands in the same relation to 289. Recognising perfect squares and their roots is a key part of number based reasoning in competitive exams.

Given Data / Assumptions:
1) The first number in the model pair is 841.
2) The second number is 29.
3) The next starting number is 289, and we need to find its partner from the options.
4) The options are 23, 21, 17, and 13.
5) We suspect a square square root or similar relation because 841 and 289 are both well known perfect squares.

Concept / Approach:
First recognise that 29 multiplied by 29 equals 841, which means 841 is 29 squared. The pattern therefore appears to be that the first number is the square of the second. If that is true for the first pair, then in the second pair 289 should be the square of the missing number. Therefore, we need to find the integer whose square is 289 and then select that integer from the options.

Step-by-Step Solution:
Step 1: Verify the relation between 841 and 29. Calculate 29 * 29. Step 2: 29 * 29 equals 841, confirming that 841 is 29 squared. Step 3: Conclude that the pattern is first number equals (second number)^2. Step 4: Apply this pattern in reverse for the second pair. We are given the first number 289 and need the second number x such that x * x equals 289. Step 5: Recognise that 17 * 17 equals 289, so the square root of 289 is 17. Step 6: Check the options and see that 17 appears as option c. Step 7: Therefore, 289 should be paired with 17 in the same way that 841 is paired with 29.

Verification / Alternative check:
Check the other options quickly. 23 * 23 equals 529, 21 * 21 equals 441, and 13 * 13 equals 169. None of these equals 289, so they cannot complete the analogy if the pattern is based on squares. Since the square pattern works perfectly for the first pair and leads uniquely to 17 for the second, the rule is consistent and reliable.

Why Other Options Are Wrong:
The numbers 23, 21, and 13 do not satisfy x squared equals 289. Choosing them would mean ignoring the clearly defined square relation in the model pair. Because a good analogy demands one single consistent rule across pairs, these options serve only as distractors.

Common Pitfalls:
Some students may misremember or miscalculate squares of two digit numbers, confusing 17 squared with another value. To avoid this, it helps to memorise squares of integers up to at least 30. Others may look for a more complicated pattern instead of noticing the direct and simple square relation. Always test basic operations such as squares and roots before searching for complex patterns.

Final Answer:
Following the same square square root pattern, 289 is related to 17 in the analogy 841 : 29 :: 289 : ?.