In the following question, four number pairs are given: 49 – 64, 576 – 729, 441 – 484 and 100 – 121. In three of these pairs, the numbers are perfect squares of consecutive integers, while one pair does not follow this consecutive-square relationship. Which pair is the odd one out?

Introduction / Context:
This question is based on recognising perfect squares and patterns between them. You are given four number pairs and must determine which pair does not follow the same relationship as the others. Specifically, three pairs consist of perfect squares of consecutive integers, while one pair has a different gap between the square roots of the two numbers. Such questions help test your familiarity with square numbers and your ability to detect simple numerical patterns.

Given Data / Assumptions:

• Number pairs: 49 – 64, 576 – 729, 441 – 484, 100 – 121.
• We will express each number as a square of some integer.
• We then examine the difference between the square roots in each pair.
• Three pairs are expected to have consecutive integer square roots, and one pair will not.

Concept / Approach:
The method is straightforward:

• Write each number as n^2 if possible.
• For each pair (a, b), find integers m and n such that a = m^2 and b = n^2.
• Check whether n = m + 1, meaning that the second number is the square of the next consecutive integer.

The pair where this relationship does not hold is the odd one out.

Step-by-Step Solution:
Step 1: For 49 – 64: 49 = 7^2 and 64 = 8^2. The square roots are 7 and 8, which are consecutive integers. Step 2: For 441 – 484: 441 = 21^2 and 484 = 22^2. The square roots are 21 and 22, again consecutive integers. Step 3: For 100 – 121: 100 = 10^2 and 121 = 11^2. The square roots are 10 and 11, which are consecutive. Step 4: For 576 – 729: 576 = 24^2 and 729 = 27^2. The square roots are 24 and 27, which are not consecutive (there is 25 and 26 in between). Step 5: Therefore, in three pairs the square roots differ by 1, and in one pair they differ by 3. The pair 576 – 729 breaks the consecutive square pattern.

Verification / Alternative check:
You can also check the differences between the square roots: 49 – 64: roots 7 and 8, difference = 1. 441 – 484: roots 21 and 22, difference = 1. 100 – 121: roots 10 and 11, difference = 1. 576 – 729: roots 24 and 27, difference = 3. Since only the pair 576 – 729 shows a larger gap between the square roots, it is clearly distinct from the others.

Why Other Options Are Wrong:
49 – 64 is not odd because it is formed by squares of consecutive integers 7 and 8. 441 – 484 is not odd because it uses consecutive roots 21 and 22. 100 – 121 is not odd because its roots 10 and 11 are consecutive. Only 576 – 729 uses roots 24 and 27, which are not consecutive, making it different.

Common Pitfalls:
A common mistake is to look only at the numeric differences between the pair members instead of focusing on square roots. Another error is misremembering squares such as 24^2 and 27^2. To avoid confusion, carefully compute or recall key squares and focus on the relationship between their roots, not just the raw numbers.

Final Answer:
The pair that does not consist of squares of consecutive integers and is therefore the odd one out is 576 – 729.