Question 1070285
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You got this product   {{{ (( (n-1)n(n+1)(n+2)) /12) }}}.


The numerator is the product of 4 consecutive integers,  isn't it ?


One of them is a multiple of 4.  One of them is a multiple of 3.


Therefore,  the numerator is a multiple of  12,


so the fraction is an integer number.


Proved.



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MOREOVER, it is a multiple of 24.