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If m and k are positive integers, is m!+8k a multiple of k?

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

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New post 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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New post 23 Jun 2014, 07:27
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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.

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Hope 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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New post 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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