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If 6x=8y=14z, then what is a possible sum of positive integers x, y, a

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Bunuel
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If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   19 Jan 2015, 05:22
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If 6x = 8y = 14z, then what is a possible sum of positive integers x, y, and z?

A. 52
B. 58
C. 84
D. 122
E. 168
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D
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KarishmaB
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   02 Feb 2016, 23:07
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Bunuel wrote:
If 6x = 8y = 14z, then what is a possible sum of positive integers x, y, and z?

A. 52
B. 58
C. 84
D. 122
E. 168

Kudos for a correct solution.


A rule of thumb for 2 or more "=" signs in a single equation: Put them further equal to k and bring everything in terms of k.

6x = 8y = 14z = k

x = k/6
y = k/8
z = k/14

For x, y and z to be integers, k must be divisible by 6, 8 as well as 14. The LCM of 6, 8 and 14 is 6*4*7 = 168. So k must be 168 or a multiple of it.

x = 168a/6 = 28a
y = 168a/8 = 21a
z = 168a/14 = 12a

x + y + z = 28a + 21a + 12a = 61a

The only multiple of 61 we have is 122. Answer (D)
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   20 Jan 2015, 03:06
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Answer = D = 122

6x = 8y = 14z

3x = 4y = 7z

3(4*7) = 4(3*7) = 7(3*4)

Addition = 28+21+12 = 61

Answer would be multiple of 61 which is 122
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   19 Jan 2015, 05:53
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Bunuel wrote:
If 6x = 8y = 14z, then what is a possible sum of positive integers x, y, and z?

A. 52
B. 58
C. 84
D. 122
E. 168

Kudos for a correct solution.


The possible value for which 6x = 8y =14z ,will be LCM of 6,8,14 i.e. 168
Therefore x = 28, y = 21, z = 12
sum = 61.
But we don't have 61 in the ans choices. Only possible values which satisfy the eqn given will be multiples of 168.
We move onto the next multiple(336), which also doubles the value of x, y and z.
Now x+y+z = 122, which is present in the answer choice.
Hence D.

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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   19 Jan 2015, 06:08
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Bunuel wrote:
If 6x = 8y = 14z, then what is a possible sum of positive integers x, y, and z?

A. 52
B. 58
C. 84
D. 122
E. 168

Kudos for a correct solution.



x : y
8 : 6

y : z
14 : 8

x : y : z
56 : 42 : 24

x+y+z = 122 could be possible sum
IMO D
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   27 Mar 2017, 17:26
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Bunuel wrote:
If 6x = 8y = 14z, then what is a possible sum of positive integers x, y, and z?

A. 52
B. 58
C. 84
D. 122
E. 168


We can divide the given equation by 2 and we have:

3x = 4y = 7z

In order for each term to be equal, x could be 4(7) = 28, y could be 3(7) = 21, and z could be 3(4) = 12.

Thus, we see that the smallest value of x + y + z is 28 + 21 + 12 = 61. However, since that is not an answer choice, the next highest value is 2 x 61 = 122.

Answer: D
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   19 Dec 2022, 10:09
6x = 8y = 14z → x = 14z/6 = 7z/3 and y = 14z/8 = 7z/4
Sum = x+y+z = 7z/3 + 7z/4 + z → z (7/3 + 7/4 +1) = 61z / 12

Because the sum of integers is also an integer. z must be divisible by 12.
Possible values of z = 12, 24, …

Only option that satisfies is 122 (when z = 24)
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink] New post   06 Feb 2024, 08:54
3x=4y=7z=k
Sum=k/7+k/4+k/3= 61k/84

Only option with a multiple of 61 is D)
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Re: If 6x=8y=14z, then what is a possible sum of positive integers x, y, a [#permalink]
06 Feb 2024, 08:54
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