document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=190369>Question 190369</A>: <I><SPAN STYLE=\"word-break: break-all;\"> # 5.   Find a counterexample to show that the following statement is incorrect: “The sum of any two prime numbers is divisible by 2” \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #142874 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=142874\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 5  \r<BR>\n" );
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document.write( "Q: <font color=red>Find a counterexample to show that the following statement is incorrect: “The sum of any two prime numbers is divisible by 2” </font>\r<BR>\n" );
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document.write( "A:\r<BR>\n" );
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document.write( "Recall that a prime number is a number that is ONLY divisible by 1 and itself. So the first few prime numbers are:\r<BR>\n" );
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document.write( "2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc...\r<BR>\n" );
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document.write( "Notice how every prime number (except 2) is an odd number. Now if we select two random prime numbers, there's a good chance that they'll both be odd. For example, let's say I choose 17 and 31. These are both odd numbers. If I add them, I get\r<BR>\n" );
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document.write( "17+31=48\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48%2F2=24 ALIGN=MIDDLE  ALT=\"48%2F2=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that 48 is divisible by 2. It turns out that the sum of ANY two odd numbers is ALWAYS even. So there's a good chance that the sum of any two prime numbers is even (since all of them, but the number 2, are odd)\r<BR>\n" );
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document.write( "So to find a counterexample, we MUST select 2 as one of the numbers, since it is only even prime. The other number can be odd since the sum of an even and odd number is ALWAYS odd.\r<BR>\n" );
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document.write( "So I'm going to select the numbers 2 and 3 (both are prime, but the first one is even and the second is odd)\r<BR>\n" );
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document.write( "So 2+3=5 which is NOT divisible by 2 since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2=2.5 ALIGN=MIDDLE  ALT=\"5%2F2=2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So this shows that the statement “The sum of any two prime numbers is divisible by 2” is false. We've shown this by adding the prime numbers 2 and 3 to get 5 (which is NOT divisible by 2). </SPAN>\n" );
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