document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194180>Question 1194180</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If  5x-y is divisible by 4 and 2x+3y is odd. Then 7x+2y is odd for all x,y ∈Z. Prove by direct proof or contrapositive </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826320 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=826320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "If 5x-y is divisible by 4, then it's also divisible by 2 since 4 = 2*2\r<BR>\n" );
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document.write( "Put another way:<BR>\n" );
document.write( "4 is a factor of 5x-y<BR>\n" );
document.write( "5x-y = 4k for some integer k <BR>\n" );
document.write( "5x-y = 2*(2k)<BR>\n" );
document.write( "This shows that 5x-y is even\r<BR>\n" );
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document.write( "5x-y is even while 2x+3y is odd<BR>\n" );
document.write( "(5x-y)+(2x+3y) = (5x+2x) + (-y+3y) = 7x+2y\r<BR>\n" );
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document.write( "We've summed an even number (5x-y) with an odd number (2x+3y) to get an odd result (7x+2y). This concludes the proof.\r<BR>\n" );
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document.write( "Here's a side proof showing that odd+even = odd<BR>\n" );
document.write( "This is something for another page (perhaps a reference page in your notebook somewhere).\r<BR>\n" );
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document.write( "k = some integer<BR>\n" );
document.write( "2k = some even integer<BR>\n" );
document.write( "2m+1 = some odd integer\r<BR>\n" );
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document.write( "odd+even = (2m+1)+2k = (2m+2k)+1 = 2(m+k)+1 = some other odd integer\r<BR>\n" );
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document.write( "This concludes the proof that odd+even = odd<BR>\n" );
document.write( "Since order doesn't matter when it comes to adding, it's the same as saying even+odd = odd\r<BR>\n" );
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document.write( "Some numeric examples to partially verify the claim:<BR>\n" );
document.write( "3+2 = 5<BR>\n" );
document.write( "7+14 = 21<BR>\n" );
document.write( "19+18 = 37<BR>\n" );
document.write( "These are examples to help see why the proof works. Of course you'd need the proof above to fully confirm the claim. <BR>\n" );
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