number theorygreatest common divisorpermutationscombinatorics
Problem Statement
Let a1,a2,...,an be different integers and let (b1,b2,...,bn),(c1,c2,...,cn) be two of their permutations, different from the identity. Prove that
(∣a1−b1∣+∣a2−b2∣+...+∣an−bn∣,∣a1−c1∣+∣a2−c2∣+...+∣an−cn∣)≥2
where (x,y) denotes the greatest common divisor of the numbers x,y