document.write( "Question 977111: show that square of every positive integer takes any one of the form 3p or 3p+1? \n" ); document.write( "

Algebra.Com's Answer #598724 by KMST(5328) $\"\"$ $\"About$

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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.
\n" ); document.write( "It must be one of those 3 cases; there is no other possibility.
\n" ); document.write( "So, a positive integer can be
\n" ); document.write( "a multiple of 3, of the form $\"3Q\"$ , or
\n" ); document.write( "be of the form $\"3Q%2B1\"$ (if it has a remainder of 1 when divided by 3), or
\n" ); document.write( "be of the form $\"3Q%2B2\"$ (if it has a remainder of 2 when divided by 3),
\n" ); document.write( "with $\"Q\"$ being a non-negative integer in each case.
\n" ); document.write( "The square of $\"3Q\"$ is $\"%283Q%29%5E2=%283%5E2%29%28Q%5E2%29=9Q%5E2=3%2A3Q%5E2=3p\"$ with $\"p=3Q%5E2\"$ .
\n" ); document.write( "The squares of $\"3Q%2B1\"$ and $\"3Q%2B2\"$ can be calculated as the square of a binomial with a formula proven in algebra class:
\n" ); document.write( " $\"%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2\"$ .
\n" ); document.write( "The square of $\"3Q%2B1\"$ is

with $\"p=3Q%5E2%2B2Q\"$ .
\n" ); document.write( "The square of $\"3Q%2B2\"$ is

with $\"p=3Q%5E2%2B2Q%2B1\"$ .
\n" ); document.write( "In other words, when divided by 3, the square of an integer can
\n" ); document.write( "divide evenly,
\n" ); document.write( "or leave a remainder of 1.
\n" ); document.write( "There is no other possibility. \n" ); document.write( "

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