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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if x and y are positive integers such that x = 8y + 12,
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puneetkhurana2000
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You can solve it using algebra, but i prefer to use some numbers as explained below:-
Statement 1) x = 12 u
So, for x to be a multiple of 12, we have (36,3) and (60,6) satisfy the eqn x = 8*y + 12. In these cases the HCF is 3 and 6 respectively. Not Sufficient!!!
Statement 2) y = 12 z
So, for y to be a multiple of 12, we have (12*9,12) and (12*17,24) and (12*25,36) and (12*33,48) satisfy the eqn x = 8*y + 12. In each case the HCF is 12. Sufficient!!!
Answer B.
Statement 1) x = 12 u
So, for x to be a multiple of 12, we have (36,3) and (60,6) satisfy the eqn x = 8*y + 12. In these cases the HCF is 3 and 6 respectively. Not Sufficient!!!
Statement 2) y = 12 z
So, for y to be a multiple of 12, we have (12*9,12) and (12*17,24) and (12*25,36) and (12*33,48) satisfy the eqn x = 8*y + 12. In each case the HCF is 12. Sufficient!!!
Answer B.
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GMATGuruNY
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For many test-takers, the most efficient approach on test day would be to plug in.varun289 wrote: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.
Statement 1: x=12u, where u is an integer and x=8y+12.
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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Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
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As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
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Brent@GMATPrepNow
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Target question: What is the greatest common divisor of x and y?varun289 wrote: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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alexander.vien
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GCD or Greatest Common Divisor = GCF or Greatest Common Factor
Whenever the word "divisor" is seen in a GMAT question, it can be replaced with "factor" - since factors and multiples are what most study when preparing for the GMAT.
Just wanted to clarify since this constantly confuses me!
Whenever the word "divisor" is seen in a GMAT question, it can be replaced with "factor" - since factors and multiples are what most study when preparing for the GMAT.
Just wanted to clarify since this constantly confuses me!
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Brent@GMATPrepNow
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Yes, "factor" is the same as "divisor"
In fact, there are many ways to express the concepts of divisors and factors.
For example, "y is a factor of x" can be reworded in the following ways:
"y is a divisor of x"
"x is divisible by y"
"When x is divided by y the remainder is zero"
"x equals ky for some integer k"
"x is a multiple of y"
or even, "y is hiding in the prime factorization of x"
Cheers,
Brent
In fact, there are many ways to express the concepts of divisors and factors.
For example, "y is a factor of x" can be reworded in the following ways:
"y is a divisor of x"
"x is divisible by y"
"When x is divided by y the remainder is zero"
"x equals ky for some integer k"
"x is a multiple of y"
or even, "y is hiding in the prime factorization of x"
Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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alexander.vien
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Also want to point out for this question that a quick way of knowing B is sufficient is the rule that states that if two numbers are both multiplies of a certain number, adding or subtracting them will generate an answer that is a multiple of that same number.
For example: 35 (multiple of 7) - 21 (multiple of 7) = 14 (multiple of 7)
So, we can apply this rule to statement 2.
if y = 12z, then x = 96z + 12 -> in other words, y is a multiple of 12, so a multiple of 12 added together with a multiple of 12 must be a multiple of 12! So, x is a multiple of 12 as well. Knowing that x will always be greater than y, means that 12 will always be the greatest common factor of x and y.
For example: 35 (multiple of 7) - 21 (multiple of 7) = 14 (multiple of 7)
So, we can apply this rule to statement 2.
if y = 12z, then x = 96z + 12 -> in other words, y is a multiple of 12, so a multiple of 12 added together with a multiple of 12 must be a multiple of 12! So, x is a multiple of 12 as well. Knowing that x will always be greater than y, means that 12 will always be the greatest common factor of x and y.
