Prove by mathematical induction: 6 divides n^3 - n for n>=2
Let be the proposition that for the positive integer is divisible by 6.
Basis Step:
is true, because is divisible by 6.
Inductive Step:
For the inductive hypothesis we assume that holds for an arbitrary
positive integer That is, we assume that is divisible by 6
Under this assumption, it must be shown that is true, namely, that
is divisible by 6, is also true.
If is even, then is divisible by 6.
If is odd, then is even, and is divisible by 6.
This last equation shows that is true under the assumption that is true. This completes the inductive step.
We have completed the basis step and the inductive step, so by mathematical induction we know that is true for all integers That is, we have proved that for the positive integer is divisible by 6.
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