In the number analogy "101 : 10201 :: 107 : ____", which number correctly completes the pattern by using a square?

Introduction / Context:
This analogy problem is built around squaring numbers. The pair "101 : 10201" suggests that 10201 is generated from 101 by squaring or another power based operation. To complete the analogy, you must find the number related to 107 under the same rule. Such questions are common in quantitative aptitude sections where recognition of squares and cubes is essential.

Given Data / Assumptions:

• First pair: 101 is related to 10201.
• Second pair: 107 is related to an unknown number.
• Options: 10707, 10749, 11449, 11407, 10207.
• We assume integer powers and basic multiplication operations.

Concept / Approach:
We test whether 10201 is equal to 101^2. Compute 101 * 101. Using binomial expansion, (100 + 1)^2 equals 100^2 + 2 * 100 * 1 + 1^2, which is 10000 + 200 + 1 = 10201. Hence 10201 is exactly 101^2. The pattern is therefore "number : its square". We then square 107 to get the corresponding number in the second pair.

Step-by-Step Solution:

Step 1: Confirm that 10201 is 101 squared. Compute 101^2 = (100 + 1)^2 = 10000 + 200 + 1 = 10201. Step 2: Conclude that the rule is "second number equals the square of the first number". Step 3: Apply the rule to 107. Compute 107^2 = 107 * 107. Step 4: Use the expansion (100 + 7)^2 = 10000 + 2 * 100 * 7 + 49 = 10000 + 1400 + 49. Step 5: Add to get 10000 + 1400 + 49 = 11449. Step 6: Select 11449 from the options as the correct square of 107.

Verification / Alternative check:
Check that none of the other options equals 107 squared. 10707 and 10207 look similar to the original number but are not squares. 10749 is close in value but does not result from the exact square computation. 11407 is also near but not equal to 11449. Only 11449 matches the exact arithmetic of 107 * 107. This confirms that we have identified the unique correct answer.

Why Other Options Are Wrong:
The incorrect options are designed to confuse candidates who rely on rough estimation instead of precise calculation. Since the pattern is clearly squaring, only the exact square may be accepted as correct. Any number that is not exactly equal to 107^2 does not fit the analogy and must be rejected.

Common Pitfalls:
One pitfall is to assume that a repeated or visually similar pattern like 10707 must be related, even though it is not a square. Another is to approximate the square and then select a nearby option without fully calculating. To avoid mistakes, especially when numbers are close together, you should compute squares carefully, using simple algebraic shortcuts.

Final Answer:
Following the rule "number : its square", 101 maps to 10201 and 107 maps to 11449, completing the analogy "101 : 10201 :: 107 : 11449".