document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023806>Question 1023806</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the least positive integer n such that n! is divisible by 1000?\r<BR>\n" );
document.write( "\n" );
document.write( "The sum of the first five terms of an arithmetic sequence is 90 less than the sum of the next five terms. What is the absolute difference between two consecutive terms of this sequence? Express your answer as a common fraction. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639303 by </B><A HREF=/tutors/aboutme.mpl?userid=msalonga><B>msalonga(4)</B></A><img src=\"/images/stars/iconNewId_16x16.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=msalonga><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=639303\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What is the least positive integer n such that n! is divisible by 1000?\r<BR>\n" );
document.write( "\n" );
document.write( "List down pairs where product divisible by 10 and have unique members(factorial uses unique consecutive integers):\r<BR>\n" );
document.write( "\n" );
document.write( "we need at least 3 pairs (1000 = 10^3)\r<BR>\n" );
document.write( "\n" );
document.write( "First two pair<BR>\n" );
document.write( "1,10<BR>\n" );
document.write( "2,5\r<BR>\n" );
document.write( "\n" );
document.write( "Then<BR>\n" );
document.write( "3,20\r<BR>\n" );
document.write( "\n" );
document.write( "We temporarily choose 20, test if there is another pair with a member lower than 20 and unique member, next is\r<BR>\n" );
document.write( "\n" );
document.write( "4,15\r<BR>\n" );
document.write( "\n" );
document.write( "15 is lower than 20, we choose 15, test if there is any pair with a member lower than 15 and unique members\r<BR>\n" );
document.write( "\n" );
document.write( "5,2, same as 2,5 pair<BR>\n" );
document.write( "6,5, not unique (we had 5 in 2,5 pair)<BR>\n" );
document.write( "7,10 not unique (we had 10 in 1,10 pair)<BR>\n" );
document.write( "8,5 not unique (we had 5 in 2,5 pair)<BR>\n" );
document.write( "9,10 not unique (we had 10 in 1,10 pair)<BR>\n" );
document.write( "10,1 same as 1,10 pair<BR>\n" );
document.write( "11,10 not unique (we had 10 in 1,10 pair)<BR>\n" );
document.write( "12,5 not unique (we had 5 in 2,5 pair)<BR>\n" );
document.write( "13,10 not unique (we had 10 in 1,10 pair)<BR>\n" );
document.write( "14,5 not unique (we had 5 in 2,5 pair)\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore 15 is the least positive integer n such that n! is divisible by 1000.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The sum of the first five terms of an arithmetic sequence is 90 less than the sum of the next five terms. What is the absolute difference between two consecutive terms of this sequence? Express your answer as a common fraction.\r<BR>\n" );
document.write( "\n" );
document.write( "first 5 terms, t1,t2...t5<BR>\n" );
document.write( "next 5 terms, t6,t7...t10\r<BR>\n" );
document.write( "\n" );
document.write( "using arithmetic sum Sn<BR>\n" );
document.write( "Sn = n(t1 + tn)/2\r<BR>\n" );
document.write( "\n" );
document.write( "Sum of first 5 terms:<BR>\n" );
document.write( "Sn = 5*(t1 + t5)/2\r<BR>\n" );
document.write( "\n" );
document.write( "Sum of other 5 terms<BR>\n" );
document.write( "Sn = 5*(t6 + t10)/2\r<BR>\n" );
document.write( "\n" );
document.write( "using formula for nth term, tn = t1 + d(n-1), where d = difference of consecutive terms\r<BR>\n" );
document.write( "\n" );
document.write( "t5 = t1 + d(5-1) = t1 + 4d<BR>\n" );
document.write( "t6 = t1 + d(6-1) = t1 + 5d<BR>\n" );
document.write( "t10 = t1 + d(10-1) = t1 + 9d\r<BR>\n" );
document.write( "\n" );
document.write( "Sum of first 5 terms = Sum of next 5 terms - 90<BR>\n" );
document.write( "5(t1 + t5)/2 = 5(t6 + t10)/2 - 90<BR>\n" );
document.write( "5(t1 + t1 + 4d)/2 = 5(t1 + 5d + t1 + 9d)/2 - 90<BR>\n" );
document.write( "5(2t1 + 4d)/2 = 5(2t1 + 14d)/2 - 90<BR>\n" );
document.write( "5t1 + 10d = 5t1 + 35d - 90<BR>\n" );
document.write( "90 = 35d - 10d + 5t1 - 5t1<BR>\n" );
document.write( "90 = 25d<BR>\n" );
document.write( "d = 90/25\r<BR>\n" );
document.write( "\n" );
document.write( "d = 18/5 \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );