SOLUTION: show that square of every positive integer takes any one of the form 3p or 3p+1?

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Question 977111: show that square of every positive integer takes any one of the form 3p or 3p+1?
Answer by KMST(5328) About Me  (Show Source):
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A positive integer, when divided by 3, can divide evenly, or have a remainder of 1, or have a remainder of 2.
It must be one of those 3 cases; there is no other possibility.
So, a positive integer can be
a multiple of 3, of the form 3Q , or
be of the form 3Q%2B1 (if it has a remainder of 1 when divided by 3), or
be of the form 3Q%2B2 (if it has a remainder of 2 when divided by 3),
with Q being a non-negative integer in each case.
The square of 3Q is %283Q%29%5E2=%283%5E2%29%28Q%5E2%29=9Q%5E2=3%2A3Q%5E2=3p with p=3Q%5E2 .
The squares of 3Q%2B1 and 3Q%2B2 can be calculated as the square of a binomial with a formula proven in algebra class:
%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2 .
The square of 3Q%2B1 is with p=3Q%5E2%2B2Q .
The square of 3Q%2B2 is with p=3Q%5E2%2B2Q%2B1 .
In other words, when divided by 3, the square of an integer can
divide evenly,
or leave a remainder of 1.
There is no other possibility.