hi, if (a-b)*(b-c)*(c-a)=2011 what are a,b and c? thanks  

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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This product looks like the determinant of square Vandermonde matrix:

`[[1,a,a^2],[1,b,b^2],[1,c,c^2]]`  = det(V) = (b-a)(c-a)(c-b) = (a-b)(b-c)(c-a) = 2011

Since (a-b)(b-c)(c-a) = 2011 => a`!=` b`!=` c

`[[1,a,a^2],[1,b,b^2],[1,c,c^2]] = [[1,a,a^2],[0,b-a,b^2-a^2],[1,c,c^2]] = (b-a)(c^2 - a^2) - (b^2-a^2)(c-a)=(b-a)(c-a)(c+a-b-a)`

Consider the integer divisors of 2011: 1 and 2011.

(a-b)(b-c)(c-a) = 2011 = 1*1*2011

a-b = 1 => a = 1+b

b-c = 1 => b = 1+c

c - a = 2011 => c = 2011 + 1  + 1 + c => 2013=0 false => a-b`!=` b-c`!=` 1

Since you have get three false statements, there are no integer values for a,b,c such that (a-b)(b-c)(c-a) = 2011.

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