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

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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) About Me  (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, %287n%29%5E3+=+0%5E3+=+0%28mod7%29, which simply says that the cube of any multiple of 7 is still divisible by 7.
For the 2nd class 7n + 1,
7n+%2B+1+=+-6+%28mod7%29 ==> %287n%2B1%29%5E3+=+%28-6%29%5E3+=+-216+=+1+%28mod7%29.
For the 3rd class 7n + 2,
7n+%2B+2+=+-5+%28mod7%29 ==> %287n%2B2%29%5E3+=+%28-5%29%5E3+=+-125+=+1+%28mod7%29.
For the 4nd class 7n + 3,
7n+%2B+3+=+-4+%28mod7%29 ==> %287n%2B3%29%5E3+=+%28-4%29%5E3+=+-64+=+6+%28mod7%29.
For the 5th class 7n + 4,
7n+%2B+4+=+-3+%28mod7%29 ==> %287n%2B4%29%5E3+=+%28-3%29%5E3+=+-27+=+1+%28mod7%29.
For the 6th class 7n + 5,
7n+%2B+5+=+-2+%28mod7%29 ==> %287n%2B5%29%5E3+=+%28-2%29%5E3+=+8+=+1+%28mod7%29.
For the 7th class 7n + 6,
7n+%2B+6+=+-1+%28mod7%29 ==> %287n%2B6%29%5E3+=+%28-1%29%5E3+=+-1+=+6+%28mod7%29.
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.