SOLUTION: What is the greatest whole number that MUST be a factor of the sum of any six consecutive positive odd numbers?

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Question 1189427: What is the greatest whole number that MUST be a factor of the sum of any six consecutive positive odd numbers?
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.

ANSWER.  The sum must be multiple of 12.



Proof


Let n be the "central" number in the sequence of 6 consecutive odd numbers

    (n-5), (n-3), (n-1), (n+1), (n+3), (n+5).


Then the sum is 6n, and since n is an even number n = 2m, 6n = 12m is a multiple of 12.


Since m can be any prime number, it shows that there is no other common divisor of the sums
of this kind.

Solved.



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