document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=86891>Question 86891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. What’s the smallest number that you can write with 1’s and 0’s which is divisible by 225?\r<BR>\n" );
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document.write( "2. The divisors of 216,000 add up to 792,480. What do their reciprocals add up to?\r<BR>\n" );
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document.write( "3. Find the product: 1·2·4·8·16· . . . · 2(to the 399th power)\r<BR>\n" );
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document.write( "4. All natural numbers from 1 to 101 are written in a row.  How can the signs “+” and “-” be placed between them so that the value of the resulting expression is 0?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62861 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20054)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=62861\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. What’s the smallest number that you can write with 1’s and 0’s which is divisible by 225?<BR>\n" );
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document.write( "The multiples of 225 end in 25, 50, 75, or 00.  Only those ending in 00 could<BR>\n" );
document.write( "contain only the digits 1's and 0's.  Those are the multiples of 4×225 or 900. <BR>\n" );
document.write( "So we need a multiple of 900.<BR>\n" );
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document.write( "The multiples of 900 are just the multiples of 9 with two 0's annexed.  <BR>\n" );
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document.write( "Theorem:  A positive integer is divisible by 9 if and only if the sum of its digits is a multiple of 9.  <BR>\n" );
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document.write( "So the smallest integer which contains only 1's and 0's and has sum of digits a multiple of 9 is 111111111 (9 1's). Then the smallest multiple of 900 that has only 1's and 0's is found by annexing two 0's to that.  So the answer is 11111111100.  <BR>\n" );
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document.write( "2. The divisors of 216,000 add up to 792,480. What do their reciprocals add up to? <BR>\n" );
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document.write( "Theorem:  The sum of the reciprocals of positive integer n =<BR>\n" );
document.write( "the sum of the divisors of n divided by n.<BR>\n" );
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document.write( "Let's do it with a smaller number like 12, so you can see why.<BR>\n" );
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document.write( "The divisors of 12 are 1,2,3,4,6,12.  Their sum is 28.<BR>\n" );
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document.write( "Look at the sum of the reciprocals:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B1%2F2%2B1%2F3%2B1%2F4%2B1%2F6%2B1%2F12 ALIGN=MIDDLE  ALT=\"1%2B1%2F2%2B1%2F3%2B1%2F4%2B1%2F6%2B1%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Obviously 12 is the LCD, so when we get the LCD of 12, we have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F12%2B6%2F12%2B4%2F12%2B3%2F12%2B2%2F13%2B1%2F12 ALIGN=MIDDLE  ALT=\"12%2F12%2B6%2F12%2B4%2F12%2B3%2F12%2B2%2F13%2B1%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "or<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2B6%2B4%2B3%2B2%2B1%29%2F12 ALIGN=MIDDLE  ALT=\"%2812%2B6%2B4%2B3%2B2%2B1%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The numerator is just the sum of the divisors of 12, and<BR>\n" );
document.write( "thus equals to the fraction <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28%2F12 ALIGN=MIDDLE  ALT=\"28%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces to 7/3<BR>\n" );
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document.write( "Let the set of divisors of 216,000 be S = <BR>\n" );
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document.write( "{1,2,3,4,5,6,8,9,10,12,15,16,18,20,...,216000}.  <BR>\n" );
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document.write( "if x e S then 216000/x e S. <BR>\n" );
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document.write( "Look at this sum:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1%2B1%2F2%2B1%2F3%2B1%2F4%2B1%2F5%2B1%2F6%2B1%2F8%2B1%2F9%2B1%2F10%2B1%2F12%2B1%2F15%2B1%2F16%2B1%2F18%2B1%2F20%2B...%2B1%2F216000 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br> <BR>\n" );
document.write( "If we get the LCD of 216000, we have<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=216000%2F216000%2B108000%2F216000 ALIGN=MIDDLE  ALT=\"216000%2F216000%2B108000%2F216000\"   DISABLED_style= \"max-width:90%; height:auto;\" >+...+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F216000 ALIGN=MIDDLE  ALT=\"1%2F216000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "or the fraction<BR>\n" );
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document.write( "216000+108000+...+1<BR>\n" );
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document.write( "`````216000<BR>\n" );
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document.write( "and the numerator is just the sum of the divisiors of 216000<BR>\n" );
document.write( "which we are given to be 7982480.<BR>\n" );
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document.write( "So the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=792480%2F216000 ALIGN=MIDDLE  ALT=\"792480%2F216000\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Both numerator and<BR>\n" );
document.write( "denominator can be divided by 480 to reduce that fraction to<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1651%2F450 ALIGN=MIDDLE  ALT=\"1651%2F450\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "3. Find the product: 1·2·4·8·16· . . . · 2(to the 399th power) <BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E0 ALIGN=MIDDLE  ALT=\"2%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >·<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1 ALIGN=MIDDLE  ALT=\"2%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >·<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2 ALIGN=MIDDLE  ALT=\"2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >·<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3 ALIGN=MIDDLE  ALT=\"2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >·<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E4 ALIGN=MIDDLE  ALT=\"2%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >·...·<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E399 ALIGN=MIDDLE  ALT=\"2%5E399\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "The sum of those exponents are the sum of the first 399 positive <BR>\n" );
document.write( "integers.<BR>\n" );
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document.write( "The formula for the first n positive integers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So the sum of the exponents is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28399%28400%29%29%2F2 ALIGN=MIDDLE  ALT=\"%28399%28400%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 79800,<BR>\n" );
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document.write( "making the answer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E79800 ALIGN=MIDDLE  ALT=\"2%5E79800\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "I'll come back to the last one. I haven't time just now.<BR>\n" );
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document.write( "Edwin </SPAN>\n" );
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