SOLUTION: What consecutive odd numbers. With out skipping add up to 291

Question 386944: What consecutive odd numbers. With out skipping add up to 291
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Suppose the integers are a, a+2, a+4, ..., a+2n where a is an odd integer. We have

a + (a+2) + (a+4) + ... + (a+2n) = 291

There are n+1 a's, so we can move them over:

a(n+1) + (2+4+...+2n) = 291

a(n+1) + n(n+1) = 291 (noting that 2+4+...+2n = n(n+1))

(n+1)(a+n) = 291

Since the factors of 291 are 1, 291, 3, 97, we have:

n+1 = 1, n+1 = 291, n+1 = 3, n+1 = 97

n = 0, 290, 2, 96, respectively

It follows that a = 291, -289, 95, -93

From this, we conclude that {291}, {-289, -287, ..., 291}, {95, 97, 99} and {-93, -91, ..., 99} are all sets of consecutive odd numbers that add up to 291.

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