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If x and y are positive integers, and (4x)/(3y) is an integer, which

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Bunuel
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If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   04 Jul 2023, 01:30
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If x and y are positive integers, and \(\frac{4x}{3y}\) is an integer, which of the following must be true?

I. x is a multiple of 4.
II. y is a multiple of 3.
III. x·y is a multiple of 3.

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III


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C
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ChandlerBong
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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   04 Jul 2023, 05:43
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Asked: which of the following must be true?

Given: x and y are integers and \(\frac{4x}{3y}\) = integer.

Solution: The best way to solve a MBT question is to find the values for which the statement is false.

Now we can write 4x/3y = k (Where k is a positive integer) -> 4x = 3y * k -- (1)

I. x is a multiple of 4.

x can be 3 and y can be 4, and we would get an integer. Hence, statement I is not always true.

II. y is a multiple of 3.

Similar to statement I we can use x as 3 and y as 4, we get an integer. Hence, statement II is not always true.

III. xy is a multiple of 3.

Yes, this can be true. We can take the above values of x as 3 and y as 4, and we will get xy as a multiple of 3.

We can check this for other values too, and in each case, we will get a Yes. eg. x can be 9 and y can be 12, and we will get a Yes.


Answer: C.
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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   05 Jul 2023, 01:49
Approach : Given, x and y are positive integers, and \(\frac{4x}{3y}\) is also an integer. For any fraction to be integer denominator should divide into numerator such that no remainder is left.

Stmt 1: x is a multiple of 4. If x is a multiple of 4 say 8, even then \(\frac{4x}{3y}\) is not an integer.

Stmt 2: y is a multiple of 3. If y is a multiple of 3 say 6, even then \(\frac{4x}{3y}\) is not an integer.

Stmt 3: x·y is a multiple of 3. If xy = 3k where k an integer.
Then \(\frac{4x}{3y}\) = \(\frac{4 (3k)}{3y^2 }\) and if y = 2 then condition satisfied.

Hence C is the answer.

Bunuel wrote:
If x and y are positive integers, and \(\frac{4x}{3y}\) is an integer, which of the following must be true?

I. x is a multiple of 4.
II. y is a multiple of 3.
III. x·y is a multiple of 3.

A. I only
B. II only
C. III only
D. I and II only
E. I, II and III


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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   05 Jul 2023, 02:24
Since it says "must be true", how are we saying xy must be a multiple of 3? (Say 3k)

4k/(y^2) may or may not be an integer even though 4x/3y is an integer

Waiting for your solution, Bunuel

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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   05 Jul 2023, 02:36
Hi TheBipedalHorse, surely Bunuel can throw more light on it, however i'll try to do my bit.

You are right, for certain values of k, \(4k/(y^2)\) may or may not be an integer. But observe that for certain values of k, it is an integer for sure. Same can not be said for statement 1 and 2.
In statement 1, even if x is a multiple of 4, for no value of y, \(\frac{4x}{3y}\) can be an integer.

Also for your must be true query, kindly note that none of the choice contain "None" as an option. Therefore we are sure that at least one of the three statements must be true.

Let me know if i can help further. :)
TheBipedalHorse wrote:
Since it says "must be true", how are we saying xy must be a multiple of 3? (Say 3k)

4k/(y^2) may or may not be an integer even though 4x/3y is an integer

Waiting for your solution, Bunuel

Posted from my mobile device
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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   05 Jul 2023, 03:12
walterwhite756 wrote:
Hi TheBipedalHorse, surely Bunuel can throw more light on it, however i'll try to do my bit.

You are right, for certain values of k, \(4k/(y^2)\) may or may not be an integer. But observe that for certain values of k, it is an integer for sure. Same can not be said for statement 1 and 2.
In statement 1, even if x is a multiple of 4, for no value of y, \(\frac{4x}{3y}\) can be an integer.

Also for your must be true query, kindly note that none of the choice contain "None" as an option. Therefore we are sure that at least one of the three statements must be true.

Let me know if i can help further. :)
TheBipedalHorse wrote:
Since it says "must be true", how are we saying xy must be a multiple of 3? (Say 3k)

4k/(y^2) may or may not be an integer even though 4x/3y is an integer

Waiting for your solution, Bunuel

Posted from my mobile device


Yes, thank you for explaining it. I can understand what you're trying to say. But the question empasizes on which of the options "MUST" be true, while your explanation, correct as it is, only works if the stem would have asked for "which of the statements CAN be true"
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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink] New post   05 Jul 2023, 09:11
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TheBipedalHorse, walterwhite756
Dropping my 2 cents here.

If (4x)/(3y) is an integer, x must be a multiple of 3 in order to "eliminate" the 3 in the denominator (their is no other way since x and y are positive integers).

If x is a multiple of 3, x*y is also a multiple of 3. So III must be true.
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Re: If x and y are positive integers, and (4x)/(3y) is an integer, which [#permalink]
05 Jul 2023, 09:11
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