Use mathematical induction to prove that n^{3}− n is divisible



Answered question


Use mathematical induction to prove that n3n
is divisible by 3 whenever n is a positive integer.

Answer & Explanation



Skilled2021-08-17Added 102 answers

In the inductive step we assumed that k3k is divisible by 3 which means that k3k=3M where is an integer. Now we can easily deduce P(k)P(k+1).
where Z=M+k2+k. Since Z is an integer, it is clear (k+1)3(k1) is divisible by 3 if k3k is divisible by 3. Thus concludes the proof.
Now there is a much easier proof that does not require induction, factor n3n as n(n1)(n+1)=(n1)n(n+1). Now n-1, n and n+2 are 3 consecutive integers, so one of them must be divisible by 3 since n mod 3 can take on 3 different values(0, 1 and 2) and the 3 consecutive integers each have a different value, so one of them will be 0 (mod 3). Since one of the integers in the product is divisible by 3, so is the whole product.

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