document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=357527>Question 357527</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Without using a calculator, test each of the following<BR>\n" );
document.write( "numbers for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11:<BR>\n" );
document.write( "a. 4,201,012 b. 1001<BR>\n" );
document.write( "c. 10,001 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #255290 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=255290\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) \r<BR>\n" );
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document.write( "Divisibility check for 2:\r<BR>\n" );
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document.write( "A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.\r<BR>\n" );
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document.write( "Notice how the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is an even number.\r<BR>\n" );
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document.write( "Since the last digit is one of the digits in the list above, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 2.\r<BR>\n" );
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document.write( "Divisibility check for 3:\r<BR>\n" );
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document.write( "<BR>\n" );
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document.write( "A number is divisible by 3 if the sum of the digits is divisible by 3.\r<BR>\n" );
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document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2%2B0%2B1%2B0%2B1%2B2=10 ALIGN=MIDDLE  ALT=\"4%2B2%2B0%2B1%2B0%2B1%2B2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 3, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 3.\r<BR>\n" );
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document.write( "---------------------------------\r<BR>\n" );
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document.write( "To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 4 (the last two digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 4 also.\r<BR>\n" );
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document.write( "A number is divisible by 5 if the last digit is either 0 or 5.\r<BR>\n" );
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document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 5.\r<BR>\n" );
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document.write( "A number is divisible by 6 if the number is divisible by <u>BOTH</u> 2 AND 3 (since 2*3=6).\r<BR>\n" );
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document.write( "The number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > may be divisible by 2 (see above), but it is NOT divisible by 3 (see above). So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 6 (since BOTH need to be divisible).\r<BR>\n" );
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document.write( "To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=012 ALIGN=MIDDLE  ALT=\"012\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 (the last three digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 as well.\r<BR>\n" );
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document.write( "Divisibility check for 9:\r<BR>\n" );
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document.write( "A number is divisible by 9 if the sum of the digits is divisible by 9.\r<BR>\n" );
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document.write( "\n" );
document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B2%2B0%2B1%2B0%2B1%2B2=10 ALIGN=MIDDLE  ALT=\"4%2B2%2B0%2B1%2B0%2B1%2B2=10\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 9, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 9.\r<BR>\n" );
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document.write( "A number is divisible by 10 if the last digit is a 0.\r<BR>\n" );
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document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT 0, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 10.\r<BR>\n" );
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document.write( "A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B0%2B0%2B2%3C%3E2%2B1%2B1 ALIGN=MIDDLE  ALT=\"4%2B0%2B0%2B2%3C%3E2%2B1%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 11.\r<BR>\n" );
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document.write( "So the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4201012 ALIGN=MIDDLE  ALT=\"4201012\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisble by the following numbers: 2 and 4 \r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "Divisibility check for 2:\r<BR>\n" );
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document.write( "A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.\r<BR>\n" );
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document.write( "Notice how the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is an odd number.\r<BR>\n" );
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document.write( "Since the last digit is NOT in the list above, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 2.\r<BR>\n" );
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document.write( "Divisibility check for 3:\r<BR>\n" );
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document.write( "A number is divisible by 3 if the sum of the digits is divisible by 3.\r<BR>\n" );
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document.write( "\n" );
document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0%2B0%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B0%2B0%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 3, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 3.\r<BR>\n" );
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document.write( "To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=01 ALIGN=MIDDLE  ALT=\"01\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 4 (the last two digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 4 as well.\r<BR>\n" );
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document.write( "A number is divisible by 5 if the last digit is either 0 or 5.\r<BR>\n" );
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document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 5.\r<BR>\n" );
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document.write( "A number is divisible by 6 if the number is divisible by <u>BOTH</u> 2 AND 3 (since 2*3=6).\r<BR>\n" );
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document.write( "The number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is neither divisible by 2 nor 3 (see above). So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 6.\r<BR>\n" );
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document.write( "To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=001 ALIGN=MIDDLE  ALT=\"001\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 (the last three digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 as well.\r<BR>\n" );
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document.write( "Divisibility check for 9:\r<BR>\n" );
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document.write( "A number is divisible by 9 if the sum of the digits is divisible by 9.\r<BR>\n" );
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document.write( "\n" );
document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0%2B0%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B0%2B0%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 9, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 9.\r<BR>\n" );
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document.write( "A number is divisible by 10 if the last digit is a 0.\r<BR>\n" );
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document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT 0, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 10.\r<BR>\n" );
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document.write( "A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0=0%2B1 ALIGN=MIDDLE  ALT=\"1%2B0=0%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is divisible by 11.\r<BR>\n" );
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document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is only divisible by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1001 ALIGN=MIDDLE  ALT=\"1001\"   DISABLED_style= \"max-width:90%; height:auto;\" > may be divisible by other prime numbers, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the only divisible factor in our list.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "==========================================================\r<BR>\n" );
document.write( "\n" );
document.write( "c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 2:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 2 if the last digit is 2, 4, 6, 8, or 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Notice how the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is an odd number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the last digit is NOT in the list above, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 3:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 3 if the sum of the digits is divisible by 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0%2B0%2B0%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B0%2B0%2B0%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 3, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 4:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To see if a number is divisible by 4, we just need to see if the last two digits are divisible by 4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=01 ALIGN=MIDDLE  ALT=\"01\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 4 (the last two digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 4 as well.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 5:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 5 if the last digit is either 0 or 5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 5.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 6:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 6 if the number is divisible by <u>BOTH</u> 2 AND 3 (since 2*3=6).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is neither divisible by 2 nor 3 (see above). So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 6.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 8:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To see if a number is divisible by 8, we just need to see if the last three digits are divisible by 8.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=001 ALIGN=MIDDLE  ALT=\"001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 (the last three digits of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" >), this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 8 as well.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 9:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 9 if the sum of the digits is divisible by 9.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "First add up the digits in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0%2B0%2B0%2B1=2 ALIGN=MIDDLE  ALT=\"1%2B0%2B0%2B0%2B1=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Because the digits add up to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is NOT divisible by 9, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 9.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 10:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 10 if the last digit is a 0.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since the last digit of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT 0, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 10.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Divisibility check for 11:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A number is divisible by 11 if the sum of the alternating digits equals the sum of the remaining alternating digits.\r<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B0%2B1%3C%3E0%2B0 ALIGN=MIDDLE  ALT=\"1%2B0%2B1%3C%3E0%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 11.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" > is NOT divisible by 2, 3, 4, 5, 6, 8, 9, 10, or 11\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In other words, none of the numbers on the list are factors of the number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10001 ALIGN=MIDDLE  ALT=\"10001\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );