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If k, m, and t are positive integers and k/6 + m/4 = t/12

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If k, m, and t are positive integers and k/6 + m/4 = t/12 [#permalink] New post 23 Feb 2012, 01:28
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If k, m, and t are positive integers and k/6 + m/4 = t/12, do t and 12 have a common factor greater than 1?

(1) k is a multiple of 3.
(2) m is a multiple of 3.


In the explanation of this question they say that the sum of two multiples of 3 give the number that is also a multiple of 3.
Is that a general rule for any number? If someone can elaborate I would be grateful!
[Reveal] Spoiler: OA

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Re: General Math Question [#permalink] New post 23 Feb 2012, 01:40
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Stiv wrote:
If k, m, and t are positive integers and \frac {k}{6} + \frac {m}{4} = \frac {t}{12} , do t and 12 have a common factor greater than 1?
(1) k is a multiple of 3.
(2) m is a multiple of 3.


In the explanation of this question they say that the sum of two multiples of 3 give the number that is also a multiple of 3.
Is that a general rule for any number? If someone can elaborate I would be grateful!


If k, m, and t are positive integers and \frac{k}{6} + \frac{m}{4} = \frac{t}{12}, do t and 12 have a common factor greater than 1 ?

\frac{k}{6} + \frac{m}{4} = \frac{t}{12} --> 2k+3m=t.

(1) k is a multiple of 3 --> k=3x, where x is a positive integer --> 2k+3m=6x+3m=3(2x+m)=t --> t is multiple of 3, hence t and 12 have a common factor of 3>1. Sufficient.

(2) m is a multiple of 3 --> m=3y, where y is a positive integer --> 2k+3m=2k+9y=t --> t and 12 may or may not have a common factor greater than 1. Not sufficient.

Answer: A.

As for your question:
If integers a and b are both multiples of some integer k>1 (divisible by k), then their sum and difference will also be a multiple of k (divisible by k):
Example: a=6 and b=9, both divisible by 3 ---> a+b=15 and a-b=-3, again both divisible by 3.

If out of integers a and b one is a multiple of some integer k>1 and another is not, then their sum and difference will NOT be a multiple of k (divisible by k):
Example: a=6, divisible by 3 and b=5, not divisible by 3 ---> a+b=11 and a-b=1, neither is divisible by 3.

If integers a and b both are NOT multiples of some integer k>1 (divisible by k), then their sum and difference may or may not be a multiple of k (divisible by k):
Example: a=5 and b=4, neither is divisible by 3 ---> a+b=9, is divisible by 3 and a-b=1, is not divisible by 3;
OR: a=6 and b=3, neither is divisible by 5 ---> a+b=9 and a-b=3, neither is divisible by 5;
OR: a=2 and b=2, neither is divisible by 4 ---> a+b=4 and a-b=0, both are divisible by 4.

Hope it's clear.
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Re: If k, m, and t are positive integers and k/6 + m/4 = t/12 [#permalink] New post 26 Jun 2013, 01:09
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Stiv wrote:
If k, m, and t are positive integers and k/6 + m/4 = t/12, do t and 12 have a common factor greater than 1?

(1) k is a multiple of 3.
(2) m is a multiple of 3.


In the explanation of this question they say that the sum of two multiples of 3 give the number that is also a multiple of 3.
Is that a general rule for any number? If someone can elaborate I would be grateful!



We can solve the given expression and get the following

(2k+3m)/12= t/12 ------> this implies t= 2k +3 m

From St 1 we have k is a multiple of 3 so the above equation is of the form t= 2*3*a+ 3m i.e t= 6a +3m where a is a positive integer (since K is a positive integer "a" cannot be zero)

thus t = 3( 2a+m)
if a =1, m=1 then t= 9 ; an 9 and 12 have 3 as common factor other than 1
similarly if a=2, m=1 we have t=15, and both 15 and 12 have 3 as common factor
since t has 3 as one of its factors and 12 also has 3 as one of its factor and therefore "t" and 12 will always have 3 as a factor other than 1

from St2 we have t= 2k+ 3*3b -----> t= 2k+9b where b is a positive integer

Here if k=1 and b =1, then t= 11; 11 and 12 do not have any common factor other than 1
but if k=3 and b=3 then we have t= 24 ; 24 and 12 have many common factor

therefore ans should be A
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Re: If k, m, and t are positive integers and k/6 + m/4 = t/12 [#permalink] New post 26 Jun 2013, 00:39
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if k, m, and t are positive integers [#permalink] New post 22 Jan 2014, 18:21
If k, m, and t are positive integers and k/6 + m/4 = t/12 do t and 12 have a common factor greater than 1?

1. k is a multiple of 3.

2. m is a multiple of 3.


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Re: if k, m, and t are positive integers [#permalink] New post 22 Jan 2014, 18:25
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Re: if k, m, and t are positive integers   [#permalink] 22 Jan 2014, 18:25
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