Line AE: from A(0,0) to E(3,15): slope = 15/3=5, equation y=5x. Line BD: from B(8,0) to D(0,15): slope = (15-0)/(0-8) = -15/8, equation: y = (-15/8)(x-8) = (-15/8)x + 15.
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Then (x^3 + y^3 = (x+y)(x^2 - xy + y^2) = 8 \cdot (34 - 15) = 8 \cdot 19 = 152).
Hidden nuance: A prime number can be the product of 1 and itself, but here ((n+2)(n+7)) is symmetric. If one factor is prime and the other is 1, we already tried. What if one factor is -1 and the other is negative prime? That would give a positive product. Example: (n+2 = -1) → (n=-3) (no). So indeed, no positive (n) works. But the problem exists, so I must have recalled incorrectly. Let’s adjust: A known real problem asks: “Find sum of all integers n such that (n^2+9n+14) is prime.” Answer often is 0 because none exist. But competition problems avoid empty sets. Mathcounts National Sprint Round Problems And Solutions
Thus min sum = 108.
Let’s instead take a from 2018 National Sprint #22: How many positive integers (n) less than 100 have exactly 5 positive divisors? Line AE: from A(0,0) to E(3,15): slope =
Coordinates: Let A=(0,0), B=(8,0), C=(8,15), D=(0,15). E on CD: C(8,15) to D(0,15) is horizontal, so y=15. CE=5 means from C (x=8) to E (x=3) → E=(3,15).