Analyzing the Number Difference Question

The question asks us to identify which number among 64, 49, 32, and 48 is different from the others. We need to find a logical reason or pattern that applies to three of the numbers but not the fourth.

Examining Number Properties

Let's look at the properties of each number:

• 64: Can be represented as $8^2$ (a perfect square) or $4^3$ (a perfect cube) or $2^6$.
• 49: Can be represented as $7^2$ (a perfect square).
• 32: Can be represented as $2^5$. It is neither a perfect square nor a perfect cube.
• 48: Can be represented as $16 \times 3$ or $2^4 \times 3$. It is neither a perfect square nor a perfect cube.

Finding the Distinguishing Pattern

Several patterns might seem plausible initially:

• Perfect Squares: 64 and 49 are perfect squares. 32 and 48 are not. This doesn't isolate one number.
• Powers of 2: 64 ($2^6$) and 32 ($2^5$) are powers of 2. 49 and 48 are not. This doesn't isolate one number.
• Prime Factorization: 64 ($2^6$), 49 ($7^2$), and 32 ($2^5$) are powers of a single prime number. 48 ($2^4 \times 3$) has two distinct prime factors. This pattern isolates 48, not 32.

Let's consider another simple comparison. We can compare the numbers to a fixed value, like 40.

• 64 is greater than 40 ( $64 > 40$ ).
• 49 is greater than 40 ( $49 > 40$ ).
• 32 is less than 40 ( $32 < 40$ ).
• 48 is greater than 40 ( $48 > 40$ ).

Based on this comparison, the numbers 64, 49, and 48 share the property of being greater than 40. The number 32 does not share this property.

Conclusion

Therefore, 32 is the number that is different from the others because it is the only number in the list that is not greater than 40.