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# Roots: Perfect squares

If $$x$$ is an integer, is $$x$$ even? >(1) $${x}^{2}-{y}^{2}=0$$ >(2) $${x}^{2}+{y}^{2}=98$$

Stat. (1) + (2): Combine both statements. In other words, add the two equations (left to left, right to right). You get that >$$x^2 - y^2 + x^2 + y^2 = 0+98$$. The $$y$$'s reduce each other, and you're left with >$$2x^2 = 98~~~$$ _(Divide both sides by 2.)_ >$$x^2 = 49$$. Bear in mind that $$x$$ can be $$±7$$, but either way, $$x$$ is odd, and the answer is a definite "No," Hence, Statements (1) + (2) → No → S → C.

Incorrect.

**Plug In** different numbers for $$x$$ to check if you have a definite answer.

Incorrect.

**Plug In** different numbers for $$x$$ to check if you have a definite answer.

Incorrect.

**Plug In** different numbers for $$x$$ to check if you have a definite answer.

Incorrect.

**Plug In** different numbers for $$x$$ to check if you have a definite answer.

Correct.

The issue here is finding $$x$$ in order to answer the question. **Plug In** numbers to Statement (1) to see whether $$x$$ can be even or odd (e.g., Try $$x=3$$, $$y=3$$, or $$x=4$$, $$y=4$$). When the answer to a Data Sufficiency problem is sometimes "Yes" and sometimes "No," it's a "Maybe." Therefore, Statement (1) → Maybe → IS → BCE.

What about Statement (2)? Statement (2) is misleading. When you see $${x}^{2}+{y}^{2}=98$$, you instantly think of the two square numbers that would meet the equation $$49+49=98$$. This way, $$x$$ and $$y$$ are the integers 7 or -7. But are you really obliged to PI integers as $$x$$ and $$y$$? What's wrong with the equality $$64+34=98$$? Or $$1+97=98$$? Or any other sum to that extent? $$x$$ must be an integer, but there is nothing in the question stem or the statements to say that $$y$$ must be an integer as well. As long as $$x$$ is an integer (say, $$±1$$), $$y$$ can be anything needed to complete the sum to the required 98 (e.g., $$y=\sqrt{97}$$). Therefore, Statement (2) → Maybe → IS → CE.

Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient to answer the question asked.
Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient to answer the question asked.
BOTH Statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.