xsaudx Posted May 14, 2007 Share Posted May 14, 2007 If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3k is a factor of p? A. 10 B. 12 C. 14 D. 16 E. 18 Quote Link to comment Share on other sites More sharing options...
mfeaspire Posted May 18, 2007 Share Posted May 18, 2007 I think the question asks for the greatest integer k for which 3^k ,and not 3k, is a factor of p. Soln : here p = 1.2.3.....30 = 30! So the greatest integer k for which 3^k is a factor of p = greatest power of the prime factor k in 30 ! = [30/3] + [30/9] +[30/27] , where [] denotes the greatest integer function ( the rest of the terms in the series being zero) = 10 + 3 + 1 =14 Quote Link to comment Share on other sites More sharing options...
ak7 Posted May 19, 2007 Share Posted May 19, 2007 You can also think about how many different factors of 1x....x30 have 3 as a factor. These are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 Out of these, 9 and 18 contain 3 twice and 27 contains it 3 times, while the other numbers contain it once. The total number of times 3 appears during factoring is 14. Quote Link to comment Share on other sites More sharing options...
xsaudx Posted May 20, 2007 Author Share Posted May 20, 2007 thank you guys for help Quote Link to comment Share on other sites More sharing options...
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