M02-25

Official Solution:


If \(x\) and \(y\) are positive integers, is \(x\) a prime number?

(1) \(|x - 2| \lt 2 - y \).

We know that the absolute value is always non-negative, hence, the smallest possible value for \(|x-2|\) is 0. Therefore, from \(|x-2| \lt 2-y\), we get \(0 \lt 2 - y\), which simplifies to \(y \lt 2\). Combining this with the fact that \(y\) is a positive integer, we get \(y=1\).

Substituting \(y=1\) into \(|x-2| \lt 2-y\), we get \(|x-2| \lt 1\), which simplifies to \(-1 \lt x-2 \lt 1\). This gives us \(1 \lt x \lt 3\), which means \(x\) can only be 2, and 2 is a prime number. Therefore, statement 1 is sufficient to answer the question.

(2) \(x + y - 3 = |1-y|\).

Since \(y\) is a positive integer (1, 2, 3, ...), \(1-y \leq 0\). Therefore, \(|1-y| = -(1-y)\). Substituting this into the given equation, we get \(x + y - 3 = -(1-y)\), which simplifies to \(x = 2\). Since 2 is a prime number, statement 2 is also sufficient to answer the question.


Answer: D
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