document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199253>Question 1199253</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The positive integers greater than 3 are arranged in rows and columns as illustrated. In which column is the number 5018?<BR>\n" );
document.write( "https://ibb.co/G5y6tR3\r<BR>\n" );
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document.write( "There is a solution by math_helper here:<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/word/geometry/Geometry_Word_Problems.faq.question.1171707.html\r<BR>\n" );
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document.write( "But it is literally just a procedure on how to solve and makes ABSOLUTE NO SENSE TO ME OR ANY STUDENT WHO NEEDS TO 'SHOW [THEIR] WORK'. \r<BR>\n" );
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document.write( "Can someone please explain how and why this solution works, and what it is even achieving with the modulos 12?\r<BR>\n" );
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document.write( "Thanks.   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #833037 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=833037\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Mod 12 was used because the gap from 4 to 16 is +12 (when going from row1,column1 to row3,column1). Or put another way: each row has 6 items, so two such rows have 2*6 = 12 items.\r<BR>\n" );
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document.write( "The left/right pattern repeats every 12 items.\r<BR>\n" );
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document.write( "I'll take another approach.\r<BR>\n" );
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document.write( "Notice the third column has 6, 14, 18, 26, ...\r<BR>\n" );
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document.write( "We can extend that a bit to say: 6, 14, 18, 26, 30, ...<BR>\n" );
document.write( "The 30 is from writing the next two values in row 5.\r<BR>\n" );
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document.write( "Now focus on the odd numbered terms of that sequence.<BR>\n" );
document.write( "Those terms are 6, 18, 30, ...<BR>\n" );
document.write( "All of which are multiples of 6. <BR>\n" );
document.write( "Specifically they are odd multiples of 6.<BR>\n" );
document.write( "An odd multiple of 6 is of the form 6n, where n is some positive odd integer.\r<BR>\n" );
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document.write( "6*1 = 6<BR>\n" );
document.write( "6*3 = 18<BR>\n" );
document.write( "6*5 = 30<BR>\n" );
document.write( "and so on.\r<BR>\n" );
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document.write( "Another thing to notice is that the rows involving 6,18,30,... all have the terms in increasing order from left to right.<BR>\n" );
document.write( "This corresponds to odd numbered rows.\r<BR>\n" );
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document.write( "Let's divide 5018 over 6<BR>\n" );
document.write( "5018/6 = 836.333 approximately\r<BR>\n" );
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document.write( "Rounding up to the nearest odd number gets us 837<BR>\n" );
document.write( "Then,<BR>\n" );
document.write( "6*837 = 5022<BR>\n" );
document.write( "This value is in column 3 to keep up with the pattern mentioned above. This item is in row 837.\r<BR>\n" );
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document.write( "The value 5022 is 4 units over the target of 5018.<BR>\n" );
document.write( "So we have to go to the left 4 units.<BR>\n" );
document.write( "Going 2 of those units to the left gets us from column 3 to column 1.\r<BR>\n" );
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document.write( "Then we'll bump up to row 836, which will move us to column 2 in the process.<BR>\n" );
document.write( "This is the third step out of 4 total.<BR>\n" );
document.write( "The fourth final step gets us to column 3.\r<BR>\n" );
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document.write( "Here's a mini table showing what's going on<BR>\n" );
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document.write( "<tr><TD ></TD><TD >Column 1</TD><TD >Column 2</TD><TD >Column 3</TD><TD >Column 4</TD><TD >Column 5</TD><TD >Column 6</TD><TD >Column 7</TD></tr><tr><TD >Row 836</TD><TD ></TD><TD >5019</TD><TD ><font color=red>5018</font></TD><TD >5017</TD><TD >5016</TD><TD >5015</TD><TD >5014</TD></tr><tr><TD >Row 837</TD><TD >5020</TD><TD >5021</TD><TD >5022</TD><TD >5023</TD><TD >5024</TD><TD >5025</TD><TD ></TD></tr>\n" );
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document.write( "One thing I forgot to mention is that each even numbered row goes in reverse, so we'll use that fact to count down from 5019 to 5018.\r<BR>\n" );
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document.write( "Answer: <font color=red>Column 3</font><BR>\n" );
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