Question

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cho 3 số nguyên a, b, c, thỏa mãn (a - b)(b - c)(c - a)=k

CMR: (a - b)³ + (b - c)³ + (c - a)³ chia hết cho 3k

Answer 1

Ta có:

\(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3=\left(a-c\right)\left[\left(a-b\right)^2-\left(a-b\right)\left(b-c\right)+\left(b-c\right)^2\right]+\left(c-a\right)^3\) \(=\left(a-c\right)\left[\left(a-b\right)^2-\left(a-b\right)\left(b-c\right)+\left(b-c\right)^2-\left(a-c\right)^2\right]\)

\(=\left(a-c\right)\left[\left(a-b\right)\left(a-2b+c\right)+\left(b-a\right)\left(a+b-2c\right)\right]\)

\(=\left(a-c\right)\left(a-b\right)\left[\left(a-2b+c\right)-\left(a+b-2c\right)\right]\)

\(=\left(a-c\right)\left(a-b\right)\left(3c-3b\right)\)

\(=3\left(a-b\right)\left(b-c\right)\left(c-a\right)=3k\)

Vì \(3k⋮3k\) nên \(\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3⋮3k\)