document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192006>Question 1192006</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove:<BR>\n" );
document.write( "If m and n are integers, then 4 is divisible by (m + n)^2 - (m-n)^2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823893 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=823893\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "(m + n)^2 = (m + n)(m + n)<BR>\n" );
document.write( "(m + n)^2 = m(m + n) + n(m + n)<BR>\n" );
document.write( "(m + n)^2 = m^2 + mn + mn + n^2<BR>\n" );
document.write( "(m + n)^2 = m^2 + 2mn + n^2\r<BR>\n" );
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document.write( "(m - n)^2 = (m - n)(m - n)<BR>\n" );
document.write( "(m - n)^2 = m(m - n) - n(m - n)<BR>\n" );
document.write( "(m - n)^2 = m^2 - mn - mn + n^2<BR>\n" );
document.write( "(m - n)^2 = m^2 - 2mn + n^2<BR>\n" );
document.write( "Or you can use this route<BR>\n" );
document.write( "(m + n)^2 = m^2 + 2mn + n^2<BR>\n" );
document.write( "(m + (-n))^2 = m^2 + 2m(-n) + (-n)^2<BR>\n" );
document.write( "(m - n)^2 = m^2 - 2mn + n^2\r<BR>\n" );
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document.write( "Now let's subtract the two expressions<BR>\n" );
document.write( "(m + n)^2 - (m - n)^2<BR>\n" );
document.write( "(m^2 + 2mn + n^2) - (m^2 - 2mn + n^2)<BR>\n" );
document.write( "m^2 + 2mn + n^2 - m^2 + 2mn - n^2<BR>\n" );
document.write( "(m^2-m^2) + (2mn+2mn) + (n^2 - n^2)<BR>\n" );
document.write( "0m^2 + 4mn + 0n^2<BR>\n" );
document.write( "0 + 4mn + 0<BR>\n" );
document.write( "4mn\r<BR>\n" );
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document.write( "Therefore,<BR>\n" );
document.write( "(m + n)^2 - (m - n)^2 = 4mn<BR>\n" );
document.write( "is an identity for any real numbers m,n.\r<BR>\n" );
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document.write( "Since 4 is a factor of 4mn, this proves that 4mn is divisible by 4.<BR>\n" );
document.write( "Consequently, (m + n)^2 - (m - n)^2 is also divisible by 4.<BR>\n" );
document.write( "This concludes the proof.\r<BR>\n" );
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document.write( "The phrasing \"4 is divisible by (m + n)^2 - (m - n)^2\" is not correct because the order should be swapped.<BR>\n" );
document.write( "If your teacher meant \"4 is a factor of (m + n)^2 - (m - n)^2\", then it would be the correct terminology.\r<BR>\n" );
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document.write( "Another way to phrase it would be to say \"(m+n)^2 - (m-n)^2 is a multiple of 4\".<BR>\n" );
document.write( "Or you could write \"4 divides (m + n)^2 - (m - n)^2\".<BR>\n" );
document.write( "</font> </SPAN>\n" );
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