document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=9013>Question 9013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How can you prove that:\r<BR>\n" );
document.write( "\n" );
document.write( "odd + odd = even<BR>\n" );
document.write( "even + even = even (you initially put even + even = odd)<BR>\n" );
document.write( "odd + even = odd\r<BR>\n" );
document.write( "\n" );
document.write( "???? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #4886 by </B><A HREF=/tutors/aboutme.mpl?userid=prince_abubu><B>prince_abubu(198)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=prince_abubu><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=4886\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We're going to do some backwards thinking here. Hang on.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We all know that a number that ends in 0,2,4,6, or 8 is even. OK. Take ANY integer n, and you can even pick an odd number if you want. Multiply it by 2. That number n times 2 will ALWAYS turn out even. You can't help it. Now, think of it backwards: If a number A is even, it must be able to be expressed as 2*n. If A is even, then there is an integer n such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+A+=+2n+ ALIGN=MIDDLE  ALT=\"+A+=+2n+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. You can think of this trick as forcing a given variable to be even.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What about forcing a variable to be odd? Let's do this forwards first. We know that if you take any integer n and multiply it by 2, the product 2*n is ALWAYS even. Add one to that product and you will ALWAYS end up with an odd result. In other words, the quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2n+%2B+1+ ALIGN=MIDDLE  ALT=\"+2n+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is ALWAYS odd. Thinking of it backwards, you can force the number B to be odd by declaring <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+B+=+2n+%2B+1+ ALIGN=MIDDLE  ALT=\"+B+=+2n+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. AKA, there is such a number n such that the quantity <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2n+%2B+1+ ALIGN=MIDDLE  ALT=\"+2n+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" > is always undoubtedly ODD.\r<BR>\n" );
document.write( "\n" );
document.write( "---------------------------------------------------------<BR>\n" );
document.write( "Now, let's try to prove that ODD + ODD = EVEN. Let's choose two odd variables C and D. If C is odd, then C must be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2n+%2B+1+ ALIGN=MIDDLE  ALT=\"+2n+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where n is any integer. If D is odd, then D must be expressed as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2m+%2B+1+ ALIGN=MIDDLE  ALT=\"+2m+%2B+1+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Notice that we used two different variables n and m. This shows us that C and D don't have to be the same number, just as long as they're both odd. Now, let's plug it in to our equation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(2n + 1) + (2m + 1) ?= even? <---- START\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2n + 2m + 2 ?= even? <------ START No matter what n and m are, 2n and 2m will ALWAYS be even. and the 2? 2 is even, so adding an even to an even creates an even number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "odd + odd = even <----- final answer<BR>\n" );
document.write( "------------------------------------------------------------------<BR>\n" );
document.write( "If you've got two even numbers to add up, the answer is always even. Let's choose two numbers, E and F. If E and F are indeed even, then there must be integers p and q so that we can say that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=E+=+2p ALIGN=MIDDLE  ALT=\"E+=+2p\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F+=+2q ALIGN=MIDDLE  ALT=\"F+=+2q\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So let's put these to the test: 2p + 2q ?= even?\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2(p + q) ?= even? <------------- Factor out the 2 from both terms. It seems like it doesn't matter what p and q are. Their sum will be multiplied by 2, forcing the answer to be even.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "--------------------------------------------------------------------<BR>\n" );
document.write( "What about odd + even = odd? Let's choose G and H as the odd and even numbers to be added. If G is odd, then it can be written as G = 2r + 1 and r is any integer. If H is even, then it can be written as H = 2s and s is ANY integer. Let's test this:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(2r + 1) + 2s ?= odd <----------------- start\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2r + 2s + 1 ?= odd <---------------- rearrange\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2(r + s) + 1 ?= odd <-------------- factor out the 2 from the 2r and 2s. No matter what r and s are, their sum will be multiplied by 2, forcing an even number answer. Then you've got the + 1 that forces the already even answer to be odd. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );