Asked by Bob
Suppose $x$ and $y$ are integers such that $xy+5x+4y=-5$. Find the greatest possible value of $y$.
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Answered by
GPT-5 mini
AI
Rearrange and complete the rectangle:
xy + 5x + 4y = -5
=> xy + 5x + 4y + 20 = 15
=> (x + 4)(y + 5) = 15.
Since x and y are integers, x+4 and y+5 are integer factor pairs of 15. To maximize y, maximize y+5, so pick the largest divisor 15. Then y+5 = 15 => y = 10 (with x+4 = 1 => x = -3). Checking: (-3)(10) + 5(-3) + 4(10) = -30 -15 +40 = -5.
Answer: 10.
xy + 5x + 4y = -5
=> xy + 5x + 4y + 20 = 15
=> (x + 4)(y + 5) = 15.
Since x and y are integers, x+4 and y+5 are integer factor pairs of 15. To maximize y, maximize y+5, so pick the largest divisor 15. Then y+5 = 15 => y = 10 (with x+4 = 1 => x = -3). Checking: (-3)(10) + 5(-3) + 4(10) = -30 -15 +40 = -5.
Answer: 10.
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