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

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Joined: 16 Feb 2012
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If k, m, and t are positive integers and k/6 + m/4 = t/12  [#permalink]

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23 Feb 2012, 01:28
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Difficulty:

75% (hard)

Question Stats:

57% (01:25) correct 43% (01:30) wrong based on 622 sessions

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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!

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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.

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]

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26 Jun 2013, 00:39
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If k, m, and t are positive integers and k/6 + m/4 = t/12  [#permalink]

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26 Jun 2013, 01:09
1
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]

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16 Mar 2016, 05:53
Superb QUESTION
Here we need to write k as 3*p for some integer p so 3 must be in the GCD
hence A is sufficient
AS for statement 2 => t=5=> NO
for t=10=> YES
hence not sufficient
hence A
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If k, m, and t are positive integers and k/6 + m/4 = t/12  [#permalink]

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25 Nov 2018, 23:14
Could someone kindly correct or reinforce my logic below? I struggled to understand how statement (b) was incorrect, but as I typed this post out I think i've got it:

T and 12 can have either 2, 4 or 3 as a common prime factor.

If T = 2k + 3m and m=multiple of 3 then lets test values for k and m

k=0 and m=3
T= 2(0) + 3(3)
T = 9 therefore T and 12 share GCF of 3

k=1 and m=3
T= 2(1) + 3(3)
T= 2 + 9
T = 11

T and 12 share no common factors. (consecutive integer rule) GCF = 1

Since there are 2 possible answers --(b)-->IS
Bunuel
If k, m, and t are positive integers and k/6 + m/4 = t/12 &nbs [#permalink] 25 Nov 2018, 23:14
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