document.write( "Question 479798: Prove that a squared = 0 or 1 mod 3 for all a∈ Integers \n" ); document.write( "

Algebra.Com's Answer #328738 by ccs2011(207) $\"\"$ $\"About$

You can put this solution on YOUR website!
Set up 3 cases for a:
\n" ); document.write( "a = 3n
\n" ); document.write( "a = 3n+1
\n" ); document.write( "a = 3n+2
\n" ); document.write( "where n represents any integer
\n" ); document.write( "Case 1:
\n" ); document.write( " $\"a%5E2+=+%283n%29%5E2+=+9n%5E2\"$
\n" ); document.write( "Divide by 3
\n" ); document.write( " $\"9n%5E2%2F3+=+3n%5E2+%2B0\"$
\n" ); document.write( "Remainder is 0 for all n
\n" ); document.write( "Case 2:
\n" ); document.write( " $\"a%5E2+=+%283n%2B1%29%5E2+=+9n%5E2%2B6n%2B1\"$
\n" ); document.write( "Divide by 3
\n" ); document.write( " $\"%289n%5E2%2B6n%2B1%29%2F3+=+3n%5E2+%2B2n+%2B%281%2F3%29\"$
\n" ); document.write( "Remainder is 1 for all n
\n" ); document.write( "Case 3:
\n" ); document.write( " $\"a%5E2+=+%283n%2B2%29%5E2+=+9n%5E2%2B12n%2B4\"$
\n" ); document.write( "Divide by 3
\n" ); document.write( " $\"%289n%5E2%2B12n%2B4%29%2F3+=+3n%5E2+%2B4n+%2B%284%2F3%29+=+3n%5E2+%2B4n+%2B1+%2B+%281%2F3%29\"$
\n" ); document.write( "Remainder is 1 for all n
\n" ); document.write( "Therefore a^2 = 0 or 1 mod 3 for all a (integers) \n" ); document.write( "

\n" );