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 #851182 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851182\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve each problem step-by-step.\r<BR>\n" );
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document.write( "**1. Finding the 4 Integers**\r<BR>\n" );
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document.write( "* Let the 4 integers be $a, b, c, d$.<BR>\n" );
document.write( "* We are given:<BR>\n" );
document.write( "    * $a + b + c + d = 24$<BR>\n" );
document.write( "    * $a \cdot b \cdot c \cdot d = 945$<BR>\n" );
document.write( "* First, find the prime factorization of 945:<BR>\n" );
document.write( "    * $945 = 3^3 \cdot 5 \cdot 7 = 3 \cdot 3 \cdot 3 \cdot 5 \cdot 7$<BR>\n" );
document.write( "* We need to find 4 integers that multiply to 945 and add to 24.<BR>\n" );
document.write( "* Let's try different combinations:<BR>\n" );
document.write( "    * If we take 3, 5, 7, the remaining factor is 9. 3+5+7+9 = 24. 3*5*7*9 = 945.<BR>\n" );
document.write( "* Thus, the integers are 3, 5, 7, and 9.\r<BR>\n" );
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document.write( "**2. Sum of Natural Numbers Divisible by 13**\r<BR>\n" );
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document.write( "* We need to find the sum of natural numbers between 500 and 1000 that are divisible by 13.<BR>\n" );
document.write( "* The first number divisible by 13 greater than 500 is:<BR>\n" );
document.write( "    * $500 / 13 \approx 38.46$, so the first number is $39 \cdot 13 = 507$.<BR>\n" );
document.write( "* The last number divisible by 13 less than 1000 is:<BR>\n" );
document.write( "    * $1000 / 13 \approx 76.92$, so the last number is $76 \cdot 13 = 988$.<BR>\n" );
document.write( "* We have an arithmetic progression (AP) with:<BR>\n" );
document.write( "    * First term ($a_1$) = 507<BR>\n" );
document.write( "    * Common difference ($d$) = 13<BR>\n" );
document.write( "    * Last term ($a_n$) = 988<BR>\n" );
document.write( "* To find the number of terms ($n$):<BR>\n" );
document.write( "    * $a_n = a_1 + (n - 1)d$<BR>\n" );
document.write( "    * $988 = 507 + (n - 1)13$<BR>\n" );
document.write( "    * $481 = (n - 1)13$<BR>\n" );
document.write( "    * $n - 1 = 481 / 13 = 37$<BR>\n" );
document.write( "    * $n = 38$<BR>\n" );
document.write( "* To find the sum of the AP ($S_n$):<BR>\n" );
document.write( "    * $S_n = \frac{n}{2}(a_1 + a_n)$<BR>\n" );
document.write( "    * $S_{38} = \frac{38}{2}(507 + 988)$<BR>\n" );
document.write( "    * $S_{38} = 19(1495) = 28405$\r<BR>\n" );
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document.write( "**3. Consecutive Numbers in AP**\r<BR>\n" );
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document.write( "* Let the three consecutive numbers in AP be $a - d$, $a$, and $a + d$.<BR>\n" );
document.write( "* We are given:<BR>\n" );
document.write( "    * $(a - d) + a + (a + d) = 15$<BR>\n" );
document.write( "    * $(a - d)^2 + (a + d)^2 = 58$<BR>\n" );
document.write( "* From the first equation:<BR>\n" );
document.write( "    * $3a = 15$<BR>\n" );
document.write( "    * $a = 5$<BR>\n" );
document.write( "* Substitute $a = 5$ into the second equation:<BR>\n" );
document.write( "    * $(5 - d)^2 + (5 + d)^2 = 58$<BR>\n" );
document.write( "    * $25 - 10d + d^2 + 25 + 10d + d^2 = 58$<BR>\n" );
document.write( "    * $50 + 2d^2 = 58$<BR>\n" );
document.write( "    * $2d^2 = 8$<BR>\n" );
document.write( "    * $d^2 = 4$<BR>\n" );
document.write( "    * $d = \pm 2$<BR>\n" );
document.write( "* If $d = 2$, the numbers are $5 - 2$, $5$, $5 + 2$, which are 3, 5, 7.<BR>\n" );
document.write( "* If $d = -2$, the numbers are $5 - (-2)$, $5$, $5 + (-2)$, which are 7, 5, 3.\r<BR>\n" );
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document.write( "**Summary**\r<BR>\n" );
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document.write( "1.  The integers are 3, 5, 7, and 9.<BR>\n" );
document.write( "2.  The sum is 28405.<BR>\n" );
document.write( "3.  The numbers are 3, 5, 7.<BR>\n" );
document.write( " </SPAN>\n" );
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