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Bunuel
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If x and y are positive integers such that x = 8y + 12, what is the 05 Mar 2018, 01:03
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Mudit27021988
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I fail

not easy at all

I want to follow this posting.

I share the same feeling. Doing really bad with DS. How many DS 700+ are expected in exam?

Posted from my mobile device
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There is no fixed number of DS questions on the test. It varies from approximately 13 to 17.

3. Strategies and Tactics for DS Section



Hope it helps.
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If x and y are positive integers such that x = 8y + 12, what is the Updated on: 17 May 2018, 04:58
Originally posted by Nived on 17 May 2018, 04:48.
Last edited by Nived on 17 May 2018, 04:58, edited 1 time in total.
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Bunuel
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

Given: \(x=8y+12\).

(1) x = 12u, where u is an integer --> \(x=12u\) --> \(12u=8y+12\) --> \(3(u-1)=2y\) --> the only thing we know from this is that 3 is a factor of \(y\). I
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Can you please explain this Bunuel?

3(u-1)=2y

So, we know that 3 is a factor of 2y, but how can we conclude that 3 is a factor of y? Is it because 2 and 3 don't have any factor in common, except 1?
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If x and y are positive integers such that x = 8y + 12, what is the 17 May 2018, 04:54
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Bunuel
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

Given: \(x=8y+12\).

(1) x = 12u, where u is an integer --> \(x=12u\) --> \(12u=8y+12\) --> \(3(u-1)=2y\) --> the only thing we know from this is that 3 is a factor of \(y\). I
Can you please explain the Bunuel?

3(u-1)=2y

So, we know that 3 is a factor of 2y, but how can we conclude that 3 is a factor of y? Is it because 2 and 3 don't have any factor in common, except 1?
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Exactly. 2y/3 = integer (because u-1 is an integer). This to be true, y must be a multiple of 3.
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If x and y are positive integers such that x = 8y + 12, what is the Updated on: 23 Aug 2023, 00:06
Originally posted by KarishmaB on 11 Sep 2018, 23:45.
Last edited by KarishmaB on 23 Aug 2023, 00:06, edited 2 times in total.
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enigma123
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer.
(2) y = 12z, where z is an integer.

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x = 8y + 12

(1) x = 12u, where u is an integer.

If x is a multiple of 12, it means 8y is a multiple of 12. Since 8 already has three 2s, we NEED y to have a 3 but it COULD have 2s and/or other factors too.
So we cannot say what the GCD of x and y is.
Not sufficient.

(2) y = 12z, where z is an integer.
If y is a multiple of 12, x is a multiple of 12 too. So they certainly have 12 common. Let's see what else they could have common.
x is 12 more than a multiple of y so the only common factors they could have are the factors of 12. We already know that they both have 12 in them. So GCD must be 12.
(Check out this post for a discussion on this: https://anaprep.com/number-properties-s ... roperties/)
Sufficient.

Answer (B)
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If x and y are positive integers such that x = 8y + 12, what is the 11 Nov 2018, 07:57
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enigma123
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

(1) x = 12u, where u is an integer.
(2) y = 12z, where z is an integer.
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\(\left\{ \begin{gathered}\\
x,y\,\, \geqslant \,\,1\,\,{\text{ints}} \hfill \\\\
x - 8y = 12\,\,\,\,\,\left( * \right) \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,?\,\, = \,\,GCD\left( {x,y} \right)\)


\(\left( 1 \right)\,\,\,x = 12u\,\,,\,\,\,u\,\,\operatorname{int} \,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,8y = 12\left( {u - 1} \right)\)

\(\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,u = 3\,\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,\,y = 3\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {12 \cdot 3\,,\,3} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 3 \hfill \\\\
\,{\text{Take}}\,\,u = 5\,\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\,\,y = 6\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( {x,y} \right) = \left( {12 \cdot 5\,,\,6} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 6\,\, \hfill \\ \\
\end{gathered} \right.\)


\(\left( 2 \right)\,\,\,y = 12z\,\,,\,\,\,z\,\,\operatorname{int} \,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,x = 12 + 8 \cdot 12 \cdot z = 12\left( {8z + 1} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,? = 12\)

\(\left( {**} \right)\,\,\,GCD\,\,\left( {z\,,\,8z + 1} \right) = \,\,k \geqslant 1\,\,\,{\text{int}}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}\\
\,\frac{z}{k} = {\text{in}}{{\text{t}}_{\text{1}}} \hfill \\\\
\,\frac{{8z + 1}}{k} = {\operatorname{int} _2}\,\,\,\,\, \hfill \\ \\
\end{gathered} \right. \Rightarrow \,\,\,\,\,\,\,\,\frac{1}{k} = {\operatorname{int} _2} - 8\left( {\frac{z}{k}} \right) = {\operatorname{int} _2} - 8 \cdot {\operatorname{int} _1} = \operatorname{int} \,\,\,\,\,\,\,\mathop \Rightarrow \limits^{k\, \geqslant \,1\,\,\,{\text{int}}} \,\,\,\,\,k = 1\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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If x and y are positive integers such that x = 8y + 12, what is the 15 Jun 2021, 15:22
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One of the hardest official DS I have seen. I have to plug in numbers to confirm things.

(1) ->

8*1 + 12 = 20
8*2 + 12 = 28
8*3 + 12 = 36 (possible)
8*4 + 12 = 44
8*5 + 12 = 52
8*6 + 12 = 60 (possible)

GCD will be the value of y and different values are possible. Insufficient.

(2) ->

96*1 + 12 = 108
96*2 + 12 = 204
96*3 + 12 = 300

For any value of z, x and y will both be divisible by 12. Sufficient.

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If x and y are positive integers such that x = 8y + 12, what is the 15 Aug 2022, 13:00
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BrentGMATPrepNow ThatDudeKnows gmatprep ScottTargetTestPrep could you kindly share your insights on this question please? Thanks for your time in advanced.
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If x and y are positive integers such that x = 8y + 12, what is the 15 Aug 2022, 14:41
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BrentGMATPrepNow ThatDudeKnows gmatprep ScottTargetTestPrep could you kindly share your insights on this question please? Thanks for your time in advanced.
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Kimberly77

Before diving in to the question, I'll point out that while this is an official question, it's an older official question. You might find one or two questions that are this complex in the most recent OGs or Quant add-ons, but keep in mind that to get a 50, you're only expected to get half the questions of that level correct. In my opinion, the biggest takeaway here is that if you can't figure out how to wrap your head around the theory, you likely have to fall back on just trying numbers to see if you can make sense of things that way. And assuming that's the path you need to take on this question, spotting that there are two points at which we are looking at linear equations makes that job much easier. The bottom line is that when you're testing numbers, make the effort to try to spot whether there's a pattern that makes that job a little easier.

If you're trying to learn the theory, Bunuel did a good job both giving the rule and explaining how that rule is derived. I'll assume that the theory isn't clicking and you're trying to go about things by testing numbers.

Statement (1) alone:
x=12 doesn't work because that makes y=0 and we need y to be a positive integer.
x=24 doesn't work because that makes y=1.5 and we need y to be a positive integer.
x = 8y + 12 is a linear function. For each 12 that we add to x, we add 1.5 to y.
x=36, y=3 ... GCD=3.
x=48, y=4.5 ... doesn't work...y not an integer.
x=60, y=6 ... GCD=6.
We already have a GCD=3 and a GCD=6.
Does statement (1) alone give us enough information to find the GCD? No. BCE.

Statement (2) alone:
Plug y=12z into x=8y+12 to get x=8(12z)+12
x=12(8z+1)
We can see that 12 will always be a CD for x and y.
Is there a way to find a CD for z and 8z+1? (This is another linear progression where fore each 1 that we add to z, we add 8 to 8z+1.)
If z=1, 8z+1 = 9.
If z=2, 8z+1 = 17.
If z=3, 8z+1 = 25.
If z=4, 8z+1 = 33.
If z=5, 8z+1 = 41.
If z=6, 8z+1 = 49.
If z=7, 8z+1 = 57.
If z=8, 8z+1 = 65.
Never going to happen. 12 is the largest we will get.
Does statement (2) alone give us enough information to find the GCD? Yes. B.

Answer choice B.
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If x and y are positive integers such that x = 8y + 12, what is the 15 Jun 2023, 10:30
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Two consecutive numbers have no common factors except 1. (Co-Prime)

Similarly, 2 consecutive multiples of a number (say x) will have X as the GCD or HCF of the numbers.
Example: 4*8 & 4*9 will have 4 as the HCF because 8 & 9 can never have anything in common except 1.

Combining the two, z & 8z will have z as HCF. But z & 8Z+1 will only have 1 in common factors.
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If x and y are positive integers such that x = 8y + 12, what is the 24 Jul 2023, 13:37
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can you please post a list of similar questions.
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If x and y are positive integers such that x = 8y + 12, what is the 25 Jul 2023, 07:23
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can you please post a list of similar questions.
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Check below:
https://gmatclub.com/forum/m36-374931.html#p2898934
https://gmatclub.com/forum/m36-374962.html#p2899137
https://gmatclub.com/forum/m36-375207.html#p2901399

5. Divisibility/Multiples/Factors



For other topics:
ALL YOU NEED FOR QUANT ! ! !
Ultimate GMAT Quantitative Megathread

Hope it helps.
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If x and y are positive integers such that x = 8y + 12, what is the 02 Oct 2025, 05:17
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Hi Bunuel,

Can we think statement 2 as - if a and b are multiples of k and are k units apart from each other then k is greatest common divisor of a and b .
Given y= 12z
hence x =12*8*y +12 and y = 12z will have GCF of 12 ?

Thanks



Quote:
If x and y are positive integers such that x = 8y + 12, what is the greatest common divisor of x and y?

Given: x = 8y + 12.

(1) x = 12u, where u is an integer.

Substituting x = 12u into x = 8y + 12 gives 12u = 8y + 12, which leads us to 3(u-1) = 2y. The only thing we know from this is that 3 is a factor of y. Is it GCD of x and y? Not clear: if x = 36, then y = 3 and GCD(x, y) = 3 but if x = 60, then y = 6 and GCD(x, y) = 6. Two different answers. Not sufficient.

(2) y = 12z, where z is an integer.

Substituting y = 12z into x = 8y + 12 gives x = 8*12z + 12, which leads us to x = 12(8z + 1). So, we have y = 12z and x = 12(8z + 1). Now, as z and 8z + 1 do not share any common factor but 1 (8z and 8z+1 are consecutive integers and consecutive integers do not share any common factor other than 1. As 8z has all the factors of z, then z and 8z+1 also do not share any common factor other than 1), then 12 must be the GCD of x and y. Sufficient.

Answer: B.

Hope it's clear.
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Hi

Can we think statement 2 as??->??if a and b are multiples of k and are k units apart from each other then k is greatest common divisor of a and b .
Given y= 12z??
hence x =12*8*y +12 and y = 12z will have GCF of 12 ?

Thanks
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If x and y are positive integers such that x = 8y + 12, what is the 03 Oct 2025, 03:34
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imaityaroy
Hi

Can we think statement 2 as??->??if a and b are multiples of k and are k units apart from each other then k is greatest common divisor of a and b .
Given y= 12z??
hence x =12*8*y +12 and y = 12z will have GCF of 12 ?

Thanks
Show more
Good question! You're thinking about GCD properties, and your final answer for Statement 2 is correct. However, the reasoning about "k units apart" needs clarification – let me show you the precise approach.

The rule you mentioned – "if a and b are multiples of k and are k units apart, then k is the GCD" – isn't the property that applies here. What matters for Statement 2 is that both x and y are multiples of 12, not their distance apart.

Correct Approach for Statement (2):

Given: \(y = 12z\) where \(z\) is an integer

Substitute into \(x = 8y + 12\):

\(x = 8(12z) + 12 = 96z + 12 = 12(8z + 1)\)

Now observe:
  • \(x = 12(8z + 1)\) → \(x\) is a multiple of \(12\)
  • \(y = 12z\) → \(y\) is a multiple of \(12\)

Key GCD Property: If \(x = 12a\) and \(y = 12b\), then \(\text{GCD}(x, y) = 12 \times \text{GCD}(a, b)\)

Therefore:
\(\text{GCD}(x, y) = \text{GCD}(12(8z+1), 12z) = 12 \times \text{GCD}(8z+1, z)\)

Now, \(\text{GCD}(8z+1, z) = \text{GCD}(1, z) = 1\)

Why? Using the property: \(\text{GCD}(kn + r, n) = \text{GCD}(r, n)\)

Final Answer: \(\text{GCD}(x, y) = 12 \times 1 = 12\)

Statement (2) is SUFFICIENT.

What Went Wrong in Your Thinking:

You tried to apply a "distance rule" that doesn't determine GCD. The winning insight is recognizing that both numbers share 12 as a factor, then factoring it out to find if any additional common factors exist.

Practice Similar Questions:

You can practice similar questions here (you'll find a lot of OG questions) - select "LCM/GCD" under "Number Properties."
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