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# Sum of n consecutive integers

## Question

You may already know that 1+2+3+4+5=15. Let's try five other consecutive integers: 14+15+16+17+18=80. How about -1+0+1+2+3=5? We might guess that the sum of 5 consecutive integers is divisible by 5. That turns out to be true. Maybe the sum of n consecutive integers is divisible by n. That turns out not to be true: 1+2 is not divisible by 2.
Question: When is the sum of n consecutive integers divisible by n, and when is it not divisible by n?

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It certainly works if n is odd; I'm guessing it never does if n is even.

For example, 3 consecutive integers can be expressed as n-1, n, and n+1; they will always sum to 3n.

The formula for the sum of n consecutive integers is n(2A+n-1)/2 if A is the smallest of the integers. This is clearly divisible by n if and only if (2A+n-1)/2 is an integer, which is true when 2A+n-1 is an even number. This is only true when n is odd.

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It certainly works if n is odd; I'm guessing it never does if n is even.

For example, 3 consecutive integers can be expressed as n-1, n, and n+1; they will always sum to 3n.

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It certainly works if n is odd; I'm guessing it never does if n is even.

For example, 3 consecutive integers can be expressed as n-1, n, and n+1; they will always sum to 3n.

this is correct but can you prove it?

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The formula for the sum of n consecutive integers is n(2A+n-1)/2 if A is the smallest of the integers. This is clearly divisible by n if and only if (2A+n-1)/2 is an integer, which is true when 2A+n-1 is an even number. This is only true when n is odd.

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