Find the odd number from the following alternatives based on whether it is a perfect cube.

Introduction / Context:
This question focuses on recognizing perfect cubes among given numbers. Perfect powers such as squares and cubes are very important in arithmetic and frequently appear in reasoning and number system questions in competitive exams.

Given Data / Assumptions:

• The numbers given are: 512, 100, 125, and 216.
• You must choose the number that is not a perfect cube, while the others are.
• Assume cube values of small integers are known or can be recalled, such as 5^3, 6^3, 8^3, and so on.

Concept / Approach:
A perfect cube is a number that can be written as n^3, where n is an integer. The straightforward approach is to recall common cubes of small integers and check whether each given number matches one of those values. If three numbers are perfect cubes and one is not, then the non cube is the odd one out.

Step-by-Step Solution:
Step 1: Check 512. We know that 8^3 equals 8 * 8 * 8, which is 512, so 512 is a perfect cube. Step 2: Check 125. We know that 5^3 equals 5 * 5 * 5, which is 125, so 125 is a perfect cube. Step 3: Check 216. We know that 6^3 equals 6 * 6 * 6, which is 216, so 216 is also a perfect cube. Step 4: Now check 100. There is no integer n such that n^3 equals 100. Cubes near 100 are 4^3 = 64 and 5^3 = 125, so 100 falls between these and is not a perfect cube. Step 5: Therefore, 100 is the only number that is not a perfect cube, making it the odd one out.

Verification / Alternative check:
You can also confirm by prime factorization. For a number to be a perfect cube, the exponent of each prime factor must be a multiple of 3. For example, 125 is 5^3, 216 is 2^3 * 3^3, and 512 is 2^9, which is (2^3)^3. In contrast, 100 factors as 2^2 * 5^2, where the exponents are not multiples of 3. This discrepancy confirms that 100 cannot be expressed as an integer cube.

Why Other Options Are Wrong:

• 512: Equal to 8^3, so it clearly fits the perfect cube property.
• 125: Equal to 5^3, and therefore a standard perfect cube.
• 216: Equal to 6^3, which is another well known perfect cube and matches the majority pattern.

Common Pitfalls:
Some learners may confuse squares and cubes or rely only on approximate mental estimation. Confusing 10^2 and 10^3 or mixing up 4^3 and 5^3 can lead to incorrect conclusions. A systematic approach using either a memorized cube table or prime factorization avoids such mistakes and ensures accurate identification of perfect cubes.

Final Answer:
100