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Is √(x + 5) an integer? (1) x is a perfect square (2) x + 5 is divisible by 5

GMAT · 2021 · Quantitative
Question:
Is √(x + 5) an integer? (1) x is a perfect square (2) x + 5 is divisible by 5

Options

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
(D) EACH statement ALONE is sufficient
(E) Statements (1) and (2) TOGETHER are NOT sufficient
Answer: (E) Statements (1) and (2) TOGETHER are NOT sufficient

Explanation:
Statement (1): x is a perfect square (e.g., x = 4), but x + 5 = 9 is a perfect square, or x = 9, then x + 5 = 14 is not. Insufficient. Statement (2): x + 5 divisible by 5 means x + 5 = 5k. Could be 5, 10, 15, 20, 25... Only some are perfect squares. Insufficient. Together: x = 4 (perfect square), x + 5 = 9 ÷ 5 = not divisible. x = 20 (not perfect square). x needs to be perfect square and x + 5 divisible by 5. Try x = 100: x + 5 = 105 (divisible by 5), √105 not integer. Try x = 4: x + 5 = 9 (not divisible by 5). No consistent answer. Insufficient.

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