**X** and **Y** are two integers. Is **X Y**? I. $(X - Y)^a (X - Y)^b$, where *a* and *b* are natural numbers such that $a b$. II. $X^p Y^q$, where *p* and *q* are two consecutive even numbers with $q

Question

**X** and **Y** are two integers. Is **X > Y**?

I. $(X - Y)^a > (X - Y)^b$, where *a* and *b* are natural numbers such that $a > b$.

II. $X^p > Y^q$, where *p* and *q* are two consecutive even numbers with $q < p$.

Which of the following is true?

Accepted Answer

**Solution:** 1. From Statement (I): - Suppose $X - Y = -2$ (which means $X < Y$). - Let $a = 4$ and $b = 2$ (with $a > b$). - Then, $(-2)^4 = 16$ and $(-2)^2 = 4$, so $16 > 4$ holds true. - This shows the inequality in statement (I) can hold even if $X < Y$. - Hence, statement (I) alone does not guarantee $X > Y$. 2. From Statement (II): - Let $p = 4$ and $q = 2$ (consecutive even numbers with $q < p$). - Consider $X = -1$ and $Y = -2$. - Then, $X^p = (-1)^4 = 1$ and $Y^q = (-2)^2 = 4$. - Here, $1 > 4$ is false. - So, statement (II) alone does not confirm $X > Y$ either. 3. Combining both statements does not resolve the ambiguity because the counterexamples satisfy the given conditions but do not confirm $X > Y$. **Answer:** Statements (I) and (II) together are not sufficient to determine whether $X > Y$.

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