Not a good problem. My guess is this came up as an application for the fundamental theorem of arithmetic. 2 is not a factor of any of the factors, so it is not a factor of the product either. On the other hand your approach is good to prove that the product of 3 consecutive odd numbers is divisible by 3, simply by writing it out as 3(3n³+4n²-n-1) - (n-1)n(n+1). But then again, who knows? In a worst-case scenario, this problem arose when the kids were learning mathematical induction, which could also be used for the proof.
Not a good problem. My guess is this came up as an application for the fundamental theorem of arithmetic. 2 is not a factor of any of the factors, so it is not a factor of the product either. On the other hand your approach is good to prove that the product of 3 consecutive odd numbers is divisible by 3, simply by writing it out as 3(3n³+4n²-n-1) - (n-1)n(n+1). But then again, who knows? In a worst-case scenario, this problem arose when the kids were learning mathematical induction, which could also be used for the proof.