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Can't Reconcile These Two Solutions

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Intern
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Joined: 15 Apr 2017
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Can't Reconcile These Two Solutions  [#permalink]

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New post 27 Jun 2017, 21:22
I can't seem to reconcile the solutions for these two questions.

question 1:
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.

Answer B
Explanation: plug y = 12z into original equation to get x = 12(8z + 1). z and 8z+1 don't share common factors, so 12 is the GCD.

question 2:
m = 4n + 9, where n is a positive integer. What is the greatest common factor of m and n?

(1) m = 9s, where s is a positive integer.
(2) n = 4t, where t is a positive integer.

Answer A
Explanation: since m is a multiple of 9, n must also be a multiple of 9. Since both are multiples of 9 and are 9 units apart, 9 is the GCD.

Why can't we apply this second explanation to statement 1 in the first question? That is, since x is a multiple of 12, y must also be a multiple of 12. Is it something to do with how the factors of 8 are also included in 12 in the first case, but the factors of 4 aren't factors of 9 in the second case?
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Re: Can't Reconcile These Two Solutions  [#permalink]

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New post 27 Jun 2017, 22:59
Here's another one:

If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?

(1) a = 2b + 6

(2) a = 3b


Answer: A
Explanation: a and b are multiples of 6 and are 6 units apart. thus 6 is GCD.

Again, for question 1 in the previous post, why can we not say that x and y are multiples of 12, are 12 units apart, and have 12 as their GCD?
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Re: Can't Reconcile These Two Solutions  [#permalink]

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New post 30 Jun 2017, 19:12
hoopsgators wrote:
Is it something to do with how the factors of 8 are also included in 12 in the first case, but the factors of 4 aren't factors of 9 in the second case?


Yes, that's exactly what's going on! Good reasoning.

If x = 8y + 12, and x = 12u, here's what you really know:

12u = 8y + 12
3u = 2y + 3

So, y has to be a multiple of 3. For instance, y = 3, u = 3. In that case, x = 36. Notice that these numbers fit the original equation: 36 = 8(3) + 12. However, y isn't a multiple of 12.

In the other situation, you can't simplify the equation like that, since there isn't a common divisor of 4 and 9.
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Re: Can't Reconcile These Two Solutions   [#permalink] 30 Jun 2017, 19:12
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