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For integers x and y, if 91x = 8y, which of the following must be true

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For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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22 Sep 2015, 02:05
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95% (hard)

Question Stats:

38% (01:09) correct 62% (01:07) wrong based on 429 sessions

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For integers x and y, if 91x = 8y, which of the following must be true?

I. y > x
II. y/7 is an integer
III. The cube root of x is an integer

A) I only
B) II only
C) III only
D) I and II
E) II and III

Kudos for a correct solution.

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Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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22 Sep 2015, 03:04
5
Bunuel wrote:
For integers x and y, if 91x = 8y, which of the following must be true?

I. y > x
II. y/7 is an integer
III. The cube root of x is an integer

A) I only
B) II only
C) III only
D) I and II
E) II and III

Kudos for a correct solution.

Statement 1: y>x

When y=x=0, equation holds but y is not greater than x
When x=-8 and y=-91, equation again holds but x>y
NOT TRUE

Statement 2: y/7 is an integer

Since x and y are integers, 91x and 8y must also be integers.
It is given that 91x=8y
or 13*7*x = 8 y
or 13x = 8y/7
To balance the equation, y/7 must be an integer
TRUE

Statement 3: The cube root of x is an integer

x can be equal to 2*2*2*3 and for this value of x,y will be 13*7*3
So, x may or may not be a cube root.
NOT TRUE

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For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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22 Sep 2015, 03:01
4
Bunuel wrote:
For integers x and y, if 91x = 8y, which of the following must be true?

I. y > x
II. y/7 is an integer
III. The cube root of x is an integer

A) I only
B) II only
C) III only
D) I and II
E) II and III

Kudos for a correct solution.

Given 91*x=8*y

It can be equal when x = 8 or -8
and y = 91 or -91

So Statement (I) y < x if we take x = -8 and y =-91
but y > x if we take x = 8 and y = -91, so (I) is not true always

Statement (2) in any case

91/7 or -91/7 is an integer that is 13 or -13. So it is true

Statement (3) cube root of 8 or -8 is 2 and -2 respectively. So it is true always.

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Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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22 Sep 2015, 14:51
1
1)
91 * 8 = 8 * 91, y > x
91 * -8 = 8 * -91, y < x
False

2)
The valid numbers for x 0, and plus/minus multiples of 8. Therefore, the valid numbers for y are 0 and multiples of +-91, which are divisible by 3.
True

3)
Consider the case where x = 8, y = 81. The root of x is not an integer, and rooting it again does not make it an integer.
False

Math Expert
Joined: 02 Sep 2009
Posts: 50623
Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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27 Sep 2015, 10:08
1
3
Bunuel wrote:
For integers x and y, if 91x = 8y, which of the following must be true?

I. y > x
II. y/7 is an integer
III. The cube root of x is an integer

A) I only
B) II only
C) III only
D) I and II
E) II and III

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

Solution: B.

Statement 1 is not necessarily true. x could equal -8 and y could equal -91, for example, in which case the equation holds but x > y” title=”x > y”/>.

Statement 2 is true: for <img src= to equal 91x, then the prime factorization: 2*2*2*y = 13*7*x. y must then be able to account for the prime factor of 7 on the other side of the equation.

And statement 3 is not necessarily true. While x must account for the factors 2*2*2, it could also include a non-cubed factor as well. For example, x could be 2*2*2*5 and y could be 13*7*5. The equation would hold because the extra 5 is accounted for on both sides. x MUST account for 2*2*2, but need not be limited to just that, as x and y could have duplicate factors on either side. Beware the statements that look very likely to be true when you face these “must be true” problems – the GMAT is a master of misdirection.
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For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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02 Jul 2016, 12:27
Hi Bunuel,

I have one doubt in this question. Why can't y be a factor of 13. If Y is a factor of 13, it will still solve the equation. Can you please explain why y/7 will be an integer.

Thanks.
Math Expert
Joined: 02 Sep 2009
Posts: 50623
Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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03 Jul 2016, 03:08
bhartiyaayus wrote:
Hi Bunuel,

I have one doubt in this question. Why can't y be a factor of 13. If Y is a factor of 13, it will still solve the equation. Can you please explain why y/7 will be an integer.

Thanks.

91x = 8y

$$\frac{x}{y} = \frac{8}{91} = \frac{2^3}{7*13}$$.

x is a multiply of 2^3 and y is a multiple of 7*13.
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Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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25 Sep 2017, 15:40
Bunuel wrote:
For integers x and y, if 91x = 8y, which of the following must be true?

I. y > x
II. y/7 is an integer
III. The cube root of x is an integer

A) I only
B) II only
C) III only
D) I and II
E) II and III

Without actually solving the equation algebraically, we can see that x could be 8 and y could be 91, since 91(8) = 8(91). However, this is not the only possibility. We see that x could be 0 and y could be 0, since 91(0) = 8(0), or x could be -8 and y could be -91, since 91(-8) = 8(-91). In any event, we see that x is a multiple of 8 (including 0 and the negative multiples) and y is a multiple of 91 (including 0 and the negative multiples).

Let’s analyze each Roman numeral:

I. y > x

Since both x and y could be 0, y is not necessarily greater than x.

Roman numeral I is not true.

II. y/7 is an integer

Since y is a multiple of 91, y/7 is an integer.

Roman numeral II must be true.

III. The cube root of x is an integer

We’ve mentioned that x is a multiple of 8. If x = 8, then the cube root of x is an integer. However, if x = 16, then the cube root of x is not an integer.

Roman numeral III is not true.

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Re: For integers x and y, if 91x = 8y, which of the following must be true  [#permalink]

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27 Sep 2018, 02:56
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Re: For integers x and y, if 91x = 8y, which of the following must be true &nbs [#permalink] 27 Sep 2018, 02:56
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