SOLUTION: There are six pairs of integers that may be chosen from the four positive integers a < b < c < d. For each such pair, the positive difference between the two integers is calculated

Question 1162658: There are six pairs of integers that may be chosen from the four positive integers a < b < c < d. For each such pair, the positive difference between the two integers is calculated. Prove that the product of these six positive differences is divisible by 12.
Answer by ikleyn(52915) (Show Source): You can put this solution on YOUR website!
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Consider these integers moduo 4.



If they are all different modulo 4, then they are 

    p= 0 (mod4),  q= 1 (mod4), r= 2(mod4) and s= 3(mod4).



In this case, we can choose two pairs from 6 differences, that produce 2 (mod4):  r-p and s-q.



If, in opposite, they are not all different (mod4), then we have at least one pair producing difference 0(mod4).



So, in any case, we have at least one multiplier 0 (mod4).



Similar logic works for (mod3).



Of the four remainders mod3, we have at least one pair with the difference 0 (mod3).



Thus, the statement is proved.

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