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Home » where \(\gcd\) denotes the greatest common divisor. - Anadea

where \(\gcd\) denotes the greatest common divisor. - Anadea

📅 May 16, 2026 👤 scraface

May 16, 2026

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📌 Wait—perhaps the problem allows degree ≤ 3. Many contest problems phrase cubic loosely. Given the values fit a quadratic, and no higher-degree terms are forced, the minimal-degree interpolating polynomial is quadratic. Since the problem asks to find $ p(0) $, and the unique cubic polynomial (in degree ≤ 3) satisfying the values must have $ a = 0 $, we proceed with $ p(x) = 2x^2 + x $, so $ p(0) = 0 $. However, to ensure degree 3, suppose we include a zero cubic term. Then $ p(x) = 0x^3 + 2x^2 + x + 0 $, and $ p(0) = 0 $.

📌 But let's verify all values:

📌 $ p(1) = 2 + 1 = 3 $,

📌 $ p(2) = 8 + 2 = 10 $,

📌 $ p(3) = 18 + 3 = 21 $,

📌 $ p(4) = 32 + 4 = 36 $.

📌 All match. Thus, even if not strictly cubic, $ p(0) = 0 $.

📌 But perhaps we misread—maybe the problem says cubic but the data forces lower degree. Since the interpolation is unique, and the values sit on a quadratic, the answer is still $ p(0) = 0 $.

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