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If n = xy then n-1 = (x-1) + (y-1) + (x-1)(y-1), x & y need not be coprime. If there are 3 factors of n, say n = xyz then is there a general form s.t. (x-1), (y-1) and (z-1) combine to sum to (n-1) and do x, y & z have to be (pairwise) coprime? (n-1) = (x-1) + (y-1) + (z-1) + (x-1)(y-1) + (x-1)(z-1) + (y-1)(z-1) + (x-1)(y-1)(z-1)?

What about 4 factors, etc.? Where could I find a proof? Thanks!

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This seems more a question about polynomial identities than over numbers. It seems to me that you could treat x,y,z as 'variables' and all this would remain unchanged. – quid Oct 6 2011 at 15:06
$\prod\limits_{i=1}^n\left(1+X_i\right) = \sum\limits_{k=0}^n \sum\limits_{1\leq i_1<i_2<...<i_k} X_{i_1}X_{i_2}...X_{i_k}$. – darij grinberg Oct 6 2011 at 15:43

closed as not a real question by Mark Sapir, quid, Douglas Zare, Felipe Voloch, Cam McLeman Oct 6 2011 at 16:45

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