document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209946>Question 1209946</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. The sum of 4 integers is 24 and their product is 945. What are those integers?\r<BR>\n" );
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document.write( "2. Find the sum of all natural numbers between 500 and 1000 which are divisible by 13.\r<BR>\n" );
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document.write( "3. If the sum of three consecutive numbers of an AP is 15 and the sum of the squares of its 1st and 3rd terms is 58, find the numbers. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851195 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851195\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "1. The sum of 4 integers is 24 and their product is 945. What are those integers?<br><BR>\n" );
document.write( "Find that the prime factorization of 945 is 3*3*3*5*7.  That's 5 factors; we need to combine 2 of them to express 945 as the product of 4 factors.  Play with the numbers to find 3*5*7*9 = 945 and 3+5+7+9 = 24.<br><BR>\n" );
document.write( "ANSWER: 3, 5, 7, and 9<br><BR>\n" );
document.write( "2. Find the sum of all natural numbers between 500 and 1000 which are divisible by 13.<br><BR>\n" );
document.write( "500/13 = 38.46... so 13*39 = 507 is the first number we are looking for.<BR>\n" );
document.write( "1000/13 = 76.93... so 13*76 = 988 is the last one.<br><BR>\n" );
document.write( "The sum we are to find is 13*(39+40+41+...+75+76)<br><BR>\n" );
document.write( "The sum of an arithmetic sequence is (number of terms)*(average of terms).<br><BR>\n" );
document.write( "Number of terms: (76-39)+1 = 38<BR>\n" );
document.write( "Average of terms = average of first and last: (39+76)/2 = 115/2<br><BR>\n" );
document.write( "The sum of the terms is 13(38*115/2) = 13*19*115 = 28405<br><BR>\n" );
document.write( "ANSWER: 28405<br><BR>\n" );
document.write( "3. If the sum of three consecutive numbers of an AP is 15 and the sum of the squares of its 1st and 3rd terms is 58, find the numbers.<br><BR>\n" );
document.write( "Since the sum of 3 consecutive terms of an AP is 15, the middle number is 5.  Do some quick calculations to find the other two numbers.<br><BR>\n" );
document.write( "4 and 6? 4^2+6^2 = 16+36 = 52, not 58<br><BR>\n" );
document.write( "3 and 7? 3^2+7^2 = 9+49 = 58.  YES!<br><BR>\n" );
document.write( "ANSWER: 3, 5, and 7<br><BR>\n" );
document.write( "NOTE: Unlike the first problem, this problem has a relatively easy formal algebraic solution.<br><BR>\n" );
document.write( "Again starting from the fact that the middle number is 5, let the other two numbers be 5-x and 5+x.  The sum of the squares of those two numbers is 58:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%285-x%29%5E2%2B%285%2Bx%29%5E2=58 ALIGN=MIDDLE  ALT=\"%285-x%29%5E2%2B%285%2Bx%29%5E2=58\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25-10x%2Bx%5E2%2B25%2B10x%2Bx%5E2=58 ALIGN=MIDDLE  ALT=\"25-10x%2Bx%5E2%2B25%2B10x%2Bx%5E2=58\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2B2x%5E2=58 ALIGN=MIDDLE  ALT=\"50%2B2x%5E2=58\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2=8 ALIGN=MIDDLE  ALT=\"2x%5E2=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2=4 ALIGN=MIDDLE  ALT=\"x%5E2=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=2 ALIGN=MIDDLE  ALT=\"x=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And then the numbers are 5-x = 5-2 = 3, 5, and 5+x = 5+2 = 7.<br><BR>\n" );
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