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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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New post 22 Sep 2015, 03:05
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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


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[Reveal] Spoiler: OA

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

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New post 22 Sep 2015, 04:01
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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.

So IMO answer is E
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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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New post 22 Sep 2015, 04:04
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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

Answer:-B

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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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New post 22 Sep 2015, 15:51
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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

Answer B

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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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New post 27 Sep 2015, 11:08
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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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New post 02 Jul 2016, 13:27
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.

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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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New post 03 Jul 2016, 04:08
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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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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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New post 25 Sep 2017, 16:40
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.

Answer: B
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Re: For integers x and y, if 91x = 8y, which of the following must be true   [#permalink] 25 Sep 2017, 16:40
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