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.
OE B
Explain
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akash singhal
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Statement 1: x=12u, where u is an integer and x=8y+12.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
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.
If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.
If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.
Since the GCD can be different values, INSUFFICIENT.
Statement 2: y=12z, where z is an integer and x=8y+12.
In other words, y is a multiple of 12.
Since we're looking for the GCD, view x in terms of its FACTORS.
If y=12, then x = 8(12) + 12 = 12(8+1) = 12*9.
The GCD of 12 and 12*9 is 12.
If y=24, then x = 8(24) + 12 = 12(8*2 + 1) = 12*17.
The GCD of 24 and 12*17 is 12.
I'm almost convinced: the GCD is 12.
Maybe one more just to be sure:
If y=36, then x = 8(36) + 12 = 12(8*3 + 1) = 12*25.
The GCD of 36 and 12*25 is 12.
SUFFICIENT.
The correct answer is B.
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GMATinsight
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x=8y+12 = 4(2y+3)akash singhal wrote: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.
OE B
Explain
i.e. x is a Multiple of 4
Statement 1: X=12u
i.e. x is a multiple of 12
i.e. y must be a multiple of 3
but since y may be an even multiple of 3 or an odd multiple of 3 so GCD will have different values. Hence,
NOT SUFFICIENT
Statement 2: Y=12z
i.e. y must be a multiple of 3 as well 4
for such value of y, x must be a multiple of 12
e.g. @y=12, x = 4*27, GCD = 12
@y=24, x = 4*51, GCD = 12
but since y is an even multiple of 3 so GCD will have constant value. Hence,
SUFFICIENT
Answer: Option B
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Brent@GMATPrepNow
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Target question: What is the greatest common divisor of x and y?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
Given: x = 8y + 12
Statement 1: x = 12u, where u is an integer.
There are several pairs of values that satisfy the given conditions. Here are two:
Case a: x=36 and y=3, in which case the GCD of x and y is 3
Case b: x=60 and y=6, in which case the GCD of x and y is 6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: y = 12z, where z is an integer.
If y = 12z and x = 8y + 12, then we can replace y with 12z to get:
x = 8(12z) + 12, which means x = 96z + 12, which means x = 12(8z + 1) [if we factor]
So, what is the GCD of 12z and 12(8z + 1)?
Well, we can see that they both share 12 as a common divisor, but what about z and 8z+1?
Well, there's a nice rule that says: The GCD of n and kn+1 is always 1 (if n and k are positive integers)
So, the GCD of z and 8z+1 is 1, which means the GCD of 12z and 12(8z + 1) is 12.
This means that the GCD of x and y is 12
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = B
Cheers,
Brent
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Amrabdelnaby
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Hi Brent,
I actually solved it using algebra. Pls tell me if my thread of thoughts is right.
Statement 1: x = 12U
12U = 8y + 12 ---> divide by 4
3U = 2y + 3
Insufficient because we have 2 unknown variables and we can't get a common factor for them
Statement 2: y = 12z
x = 8(12z) + 12
x = 12(8z + 1)
here the highest common factor is 12 so this statement is sufficient.
Is this correct thinking?
I actually solved it using algebra. Pls tell me if my thread of thoughts is right.
Statement 1: x = 12U
12U = 8y + 12 ---> divide by 4
3U = 2y + 3
Insufficient because we have 2 unknown variables and we can't get a common factor for them
Statement 2: y = 12z
x = 8(12z) + 12
x = 12(8z + 1)
here the highest common factor is 12 so this statement is sufficient.
Is this correct thinking?
GMATGuruNY wrote:Statement 1: x=12u, where u is an integer and x=8y+12.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
In other words, x is a multiple of 12.
For x to be a multiple of 12, 8y must be a multiple of 12.
If y=3, then x = 8*3 + 12 = 36.
The GCD of 3 and 36 is 3.
If y=6, then x = 8*6 + 12 = 60.
The GCD of 6 and 60 is 6.
Since the GCD can be different values, INSUFFICIENT.
Statement 2: y=12z, where z is an integer and x=8y+12.
In other words, y is a multiple of 12.
Since we're looking for the GCD, view x in terms of its FACTORS.
If y=12, then x = 8(12) + 12 = 12(8+1) = 12*9.
The GCD of 12 and 12*9 is 12.
If y=24, then x = 8(24) + 12 = 12(8*2 + 1) = 12*17.
The GCD of 24 and 12*17 is 12.
I'm almost convinced: the GCD is 12.
Maybe one more just to be sure:
If y=36, then x = 8(36) + 12 = 12(8*3 + 1) = 12*25.
The GCD of 36 and 12*25 is 12.
SUFFICIENT.
The correct answer is B.
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Matt@VeritasPrep
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The logic applied to the first statement is dangerous. For instance, suppose that I tell you that p and q are distinct primes, and that m = 2p and n = 4q. From this, we know that the GCF of m and n = 2, so we could solve a similar problem.Amrabdelnaby wrote:Hi Brent,
I actually solved it using algebra. Pls tell me if my thread of thoughts is right.
Statement 1: x = 12U
12U = 8y + 12 ---> divide by 4
3U = 2y + 3
Insufficient because we have 2 unknown variables and we can't get a common factor for them
Statement 2: y = 12z
x = 8(12z) + 12
x = 12(8z + 1)
here the highest common factor is 12 so this statement is sufficient.
Is this correct thinking?
"Two variables, two equations" is one of the more dangerous justifications to use on any GMAT problem: the test writers know that students overapply and/or misunderstand this and design many problems to trap you. Know what it does (= it tells you that, given two independent linear equations in x and y, you can uniquely determine x and y) and what it doesn't (= a zillion other things).
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Matt@VeritasPrep
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That said, we could use algebra to approach the first statement.
Since x = 8y + 12, we know that GCF(x,y) is the same as GCF(8y+12, y). Since the GCF is a factor of y, it must be a factor of 8y. Since it's a factor of 8y and of 8y + 12, it must also be a factor of their difference, or 12. So whatever the GCF is, it's a factor of 12. But the value of x itself doesn't tell us which factor of 12 this must be, so we can't solve.
Since x = 8y + 12, we know that GCF(x,y) is the same as GCF(8y+12, y). Since the GCF is a factor of y, it must be a factor of 8y. Since it's a factor of 8y and of 8y + 12, it must also be a factor of their difference, or 12. So whatever the GCF is, it's a factor of 12. But the value of x itself doesn't tell us which factor of 12 this must be, so we can't solve.
