SOLUTION: how will I show that the cube of any number is in the form 7k or 7k+1 or 7k-1

Question 462731: how will I show that the cube of any number is in the form 7k or 7k+1 or 7k-1
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
Every integer falls into one of the following modulo classes
7n
7n + 1
7n + 2
7n + 3
7n + 4
7n + 5
7n + 6
For the first class, , which simply says that the cube of any multiple of 7 is still divisible by 7.
For the 2nd class 7n + 1,
==> .
For the 3rd class 7n + 2,
==> .
For the 4nd class 7n + 3,
==> .
For the 5th class 7n + 4,
==> .
For the 6th class 7n + 5,
==> .
For the 7th class 7n + 6,
==> .
In other words, the cube of any integer is either of the form 7k, 7k +1, or 7k + 6. But numbers of the form 7k + 6 are the same numbers of the form 7k - 1. The proof is complete.

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