<div style="margin: 12px 0px; font-family: arial; color: #333333; background: #ffffff; border: solid 4px #e5e5e5; width: 100%; clear: left;"><div class="CM_CTB_Content_Wrap" style="margin: 0px; padding: 0px;background-color: #ffffff;"><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content.clipmarks.com/blog_embed/23d85f3b-3fe0-435d-a700-e0579cca9da0/49C061D6-BDA4-4E82-8A9C-29D9E310E716/" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511558" href="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511558" style="font-size: 11px;">www.urch.com</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511558">What is the number of Odd divisors of 20! ?</blockquote><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content8.clipmarks.com/images/clip-icon.gif" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707" href="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707" style="font-size: 11px;">www.urch.com</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707"><DIV>We have to find out Number of Odd factors of 20!. So Let us start by Expressing 20! in terms of Prime factors. Since</DIV> <BR /><DIV> N = 20 X 19 X18...................1 = 2^m + 3^n + 5^p + 7^q. Our First goal is to find the Highest Power of 2, 3, 5, 7, 11, 17, 19 which divide 20!.</DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707"><DIV> Highest Power of 2 Which will divide 20! is = 20/2 + 20/4 + 20/8 + 20/16 </DIV> <BR /><DIV> = 10 + 5 + 2 + 1 = 18. <B><U>Remember only Quotients need to be considered</U></B></DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707">the highest Power of 3, 5, 7,11,13,17,19 is 8 ,4,2,1,1,1,1 respectively.</blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707"><DIV>express 20! as</DIV> <BR /><DIV> 20! = 2^18 + 3^8 + 5^4 + 7^2 + 11^1 + 13^1 + 17^1 +19^1.</DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511707"> Total no of odd Divisors is (8+1) x ( 4+1) x ( 2+1) X (1+1) x(1+1) x(1+1)x(1+1) = 9 x 5x3x16</blockquote><div style="border-bottom: solid 1px #dcdcdc; white-space: nowrap; margin-bottom: 8px; background-color: #eeeeee ;background-image: url(http://clipmarks.com/images/source-bg.gif); background-repeat: repeat-x; height: 24px; line-height: 24px; vertical-align: middle; padding-bottom: 4px; color: #666666; font-size: 10px;" ><a href="http://clipmarks.com/clip-to-blog/" title="see clips that are hot right now"><img src="http://content9.clipmarks.com/images/clip-icon.gif" alt="" width="19" height="19" border="0" style="vertical-align: middle; margin: 0px 4px; display: inline; border: none; float:none;" /></a>clipped from <a title="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511955" href="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511955" style="font-size: 11px;">www.urch.com</a></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511955"><DIV>why the + 1 when multiplying:</DIV> <BR /><DIV> Because the 3 ^ 0 should also be counted in determining the number of factors. Similarly, the powers for other primes have a +1.</DIV></blockquote><div style="height: 2px; font-size: 2px; background: #dcdcdc; border-bottom: solid 1px #f5f5f5; margin: 2px 4px;"></div><blockquote style="text-align: left; padding: 0px 8px; margin: 4px 0px 8px 0px; background: transparent; border: none;" cite="http://www.urch.com/forums/gmat-problem-solving/78603-odd-divisors.html#post511955"><DIV>why they are multiplied:</DIV> To determine the total no. of distinct factors (ways they can be combined) that can be formed from the given set of prime factors</blockquote></div><div style="margin: 0px 6px 6px 4px;"><table style="font-size: 11px;border-spacing: 0px;padding: 0px;" cellpadding="0" cellspacing="0" width="100%"><tr><td style="background:transparent;border-width:0px;padding:0px;">&nbsp;</td><td align="right" style="background:transparent;border-width:0px;padding:0px;width:107px" width="107"><a href="http://clipmarks.com/share/49C061D6-BDA4-4E82-8A9C-29D9E310E716/blog/" title="blog or email this clip"><img src="http://content6.clipmarks.com/images/c2b-foot.png" border="0" alt="blog it" width="107" height="17" style="border-width:0px;padding:0px;margin:0px;" /></a></td></tr></table></div></div>

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