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If m and k are positive integers, is m!+8k a multiple of k? [#permalink]
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23 Jun 2014, 07:06
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If m and k are positive integers, is m! + 8k a multiple of k? (1) k < m (2) m = 3k OE Stat. (1): since k<m, m! includes k in its series of sequentially reducing factors down to 1. For example, if k=2 and m=3, then m! = 3·2·1 is a multiple of k=2. Therefore, both m! and 8k are multiples of k, and their sum MUST also be a multiple of k. Stat.(1)>Yes>S>AD.
Stat. (2): m is equal to k·some integer, and must be a multiple of k. Therefore, both m! and 8k are multiples of k, and their sum MUST also be a multiple of k. Stat.(2)>Yes>S>D.
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Re: If m and k are positive integers, is m!+8k a multiple of k? [#permalink]
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23 Jun 2014, 07:27
goodyear2013 wrote: If m and k are positive integers, is m!+8k a multiple of k? (1) k<m (2) m=3k OE Stat. (1): since k<m, m! includes k in its series of sequentially reducing factors down to 1. For example, if k=2 and m=3, then m! = 3·2·1 is a multiple of k=2. Therefore, both m! and 8k are multiples of k, and their sum MUST also be a multiple of k. Stat.(1)>Yes>S>AD.
Stat. (2): m is equal to k·some integer, and must be a multiple of k. Therefore, both m! and 8k are multiples of k, and their sum MUST also be a multiple of k. Stat.(2)>Yes>S>D. If m and k are positive integers, is m!+8k a multiple of k?First of all since 8k is a multiple of k, so for m!+8k to be a multiple of k m! must be a multiple of k. (1) k<m > m! = 1*2*...*m and since k<m, then k is somewhere in that sequence, which means that m! must be a multiple o k. Sufficient. (2) m=3k. Basically the same here: k = m/3, so k < m. Sufficient. Answer: D. Similar questions to practice: doestheintegerkhaveafactorpsuchthat1pk96091.htmlifxisanintegerdoesxhaveafactornsuchthat100670.htmlforanyintegerngreaterthan1ndenotestheproductof168575.htmldoesintegernhave2factorsxysuchthat1xyn165983.htmlifzisanintegeriszprime128732.htmlHope this helps.
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Re: If m and k are positive integers, is m!+8k a multiple of k? [#permalink]
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23 Jun 2014, 08:11
(a.) lets take first statement : k<m given m!+8k, we now factorial is multiplication of all the number equal to and less than the number upto 1 . so at some point if m is greater than k then its expansion will contain k at some point in expansion . so if k<m => m! = m*_*_*k*_*....1
m!+8k will have common element as k which can be taken common from whole expression and hence divisible by k.
So, (a) is SUFFICIENT to answer the question . (b) m=3k m!+8k => (3k)!+8k this is clearly divisible by k for all positive value of k and since factorial cannot negative so it always holds true. So, (b) is also SUFFICIENT to answer the question .
So answer will be either are sufficient which is (D)



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Re: If m and k are positive integers, is m!+8k a multiple of k? [#permalink]
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Re: If m and k are positive integers, is m!+8k a multiple of k?
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