Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Integer and Divisibility.
Every integer of form \((n^{3}-n)(n-2)\) for n=3,4,..... is
Logic
Integers
Divisibility
Answer: divisible by 6 but not always divisible by 12
B.Stat Objective Question 69
Challenges and Thrills of Pre-College Mathematics by University Press
\((n^{3}-n)(n-2)=n(n^{2}-1)(n-2)=(n-1)n(n+1)(n-2)\)
(n-1)n(n+1) is divisible by 3 and any two consecutive integers is divisible by 2 gcd(2,3)=1
then 6|(n-1)n(n+1) and minimum (n-2)=1 for n=3,4,.... then \((n^{3}-n)(n-2)\) divisible by 6 but not always divisible by 12.
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