document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207670>Question 1207670</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let $a$ be an integer such that $0 \le a \le 10$ and $a^2 \equiv a \pmod{11}$.  If $a \neq 0,$ then find the value of $a$. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #845674 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=845674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C%3E+0 ALIGN=MIDDLE  ALT=\"a+%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can effectively \"divide\" both sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+=+a ALIGN=MIDDLE  ALT=\"a%5E2+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" > by 'a' to arrive at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+1 ALIGN=MIDDLE  ALT=\"a+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "Technically it's not division that's happening, but instead multiplication of the multiplicative inverse. \r<BR>\n" );
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document.write( "Another pathway is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+=+a ALIGN=MIDDLE  ALT=\"a%5E2+=+a\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (mod 11)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+-+a+=+0 ALIGN=MIDDLE  ALT=\"a%5E2+-+a+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (mod 11)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%28a-1%29+=+0 ALIGN=MIDDLE  ALT=\"a%28a-1%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  (mod 11)\r<BR>\n" );
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document.write( "That leads to either a = 0 or a-1 = 0<BR>\n" );
document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%3C%3E+0 ALIGN=MIDDLE  ALT=\"a%3C%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is specified in the instructions, so we must conclude that a-1 = 0 leads to a = 1.\r<BR>\n" );
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document.write( "Yet another alternative method is to simply brute force your way through things. <BR>\n" );
document.write( "Try a = 1, a = 2, etc all the way up to a = 10.\r<BR>\n" );
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document.write( "a = 1<BR>\n" );
document.write( "a^2 = a (mod 11)<BR>\n" );
document.write( "1^1 = 1 (mod 11)<BR>\n" );
document.write( "1 = 1 (mod 11)<BR>\n" );
document.write( "That works out\r<BR>\n" );
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document.write( "a = 2<BR>\n" );
document.write( "a^2 = a (mod 11)<BR>\n" );
document.write( "2^2 = 2 (mod 11)<BR>\n" );
document.write( "4 = 2 (mod 11)<BR>\n" );
document.write( "That doesn't work out since the two sides don't boil down to same number, and we cannot reduce either side.\r<BR>\n" );
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document.write( "a = 3<BR>\n" );
document.write( "a^2 = a (mod 11)<BR>\n" );
document.write( "3^2 = 3 (mod 11)<BR>\n" );
document.write( "9 = 3 (mod 11)<BR>\n" );
document.write( "That doesn't work out either.\r<BR>\n" );
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document.write( "a = 4<BR>\n" );
document.write( "a^2 = a (mod 11)<BR>\n" );
document.write( "4^2 = 4 (mod 11)<BR>\n" );
document.write( "16 = 4 (mod 11)<BR>\n" );
document.write( "5 = 4 (mod 11)<BR>\n" );
document.write( "This is the first time that we reduce mod 11 on the left hand side. <BR>\n" );
document.write( "You can repeatedly subtract 11 from the item until getting in the range of between 0 and 10, or you can divide by 11 to look at the remainder.<BR>\n" );
document.write( "Like the previous cases, the two sides don't match, so a = 4 isn't a solution.\r<BR>\n" );
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document.write( "Keep this process going and you'll find that the values a = 5 through a = 10 are also non-solutions.\r<BR>\n" );
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document.write( "This confirms that only a = 1 is the solution.<BR>\n" );
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