document.write( "Question 755184: Prove that if n is any integer not divisible by 5, then n has a square that is either of the form 5k+1 or 5k+4, where k is some integer. \n" ); document.write( "
Algebra.Com's Answer #459528 by KMST(5328)\"\" \"About 
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If n is any integer not divisible by 5, when we divide it by 5 we get 1, 2, 3, or 4 as a remainder. That means that
\n" ); document.write( "\"n=5p%2B1\" (remainder=1), or
\n" ); document.write( "\"n=5p%2B2\" (remainder=2), or
\n" ); document.write( "\"n=5p%2B3\" (remainder=3), or
\n" ); document.write( "\"n=5p%2B4\" (remainder=4),
\n" ); document.write( "for some integer p.
\n" ); document.write( "
\n" ); document.write( "If \"n=5p%2B1\", then
\n" ); document.write( "\"n%5E2=%285p%2B1%29%5E2=25p%5E2%2B10p%2B1=5%285p%5E2%2B2p%29%2B1\",
\n" ); document.write( "which is \"n%5E2=5k%2B1\" with \"k=5p%5E2%2B2p\"
\n" ); document.write( "If \"n=5p%2B2\", then
\n" ); document.write( "\"n%5E2=%285p%2B2%29%5E2=25p%5E2%2B20p%2B4=5%285p%5E2%2B4p%29%2B4\",
\n" ); document.write( "which is \"n%5E2=5k%2B4\" with \"k=5p%5E2%2B4p\"
\n" ); document.write( "If \"n=5p%2B3\", then
\n" ); document.write( "\"n%5E2=%285p%2B3%29%5E2=25p%5E2%2B30p%2B9=25p%5E2%2B30p%2B5%2B4=5%285p%5E2%2B6p%2B1%29%2B4\",
\n" ); document.write( "which is \"n%5E2=5k%2B4\" with \"k=5p%5E2%2B6p%2B1\"
\n" ); document.write( "If \"n=5p%2B4\", then
\n" ); document.write( "\"n%5E2=%285p%2B4%29%5E2=25p%5E2%2B40p%2B16=25p%5E2%2B40p%2B15%2B1=5%285p%5E2%2B8p%2B3%29%2B1\",
\n" ); document.write( "which is \"n%5E2=5k%2B1\" with \"k=5p%5E2%2B8p%2B3\" \n" ); document.write( "
\n" );