In x and y are positive integers such that x = 8y + 12

[jnelson0612]

can someone tell me how to know 8z+1 and z are mutual and only have common factor .1 . THANK YOU!

Plug in some actual numbers for z and see if you can answer your own question. If not please return for more help.

[shadangi]

I'm not sure if this is correct, can someone please verify:

For Statement (2), we have:

y = 12Z

X = 8 x 12Z + 12 (substituting back into the given equation)

X = 12 (8Z + 1) --> Therefore, X is even

x = 12 x ODD

Given Y = 12Z

Is Z odd?

8 x 12Z = X -12 (using our substituted equation)

12 Z = (X - 12) / 8

Therefore Z = (X - 12) / 96 --> Even / Even = E (Since we know X is even) [1]

Hence Z is even. X and Y therefore don't share other common factors (one is Even --> Z and the other Odd) [2]. So 12 is the GCD.

Are you sure [1] and [2] are correct?
[1]: 6/2 != EVEN
[2]: 12*3 and 12*6 have GCD 12*3
Am I missing something?

Anyhow, I think, the right thought process for this problem is to substitute the statements' given info into original given info.

Hence, according to Stmt (2),
y = 12z
x = 12 (8z + 1)
Now what? Can we factor anything out of z and 8z+1 to make the GCD 12+ in some way?

Here is the kicker: If A is a number with p1 as "smallest" prime factor, then the "next" number after A (on the increasing number line) that has at-least one factor (other than 1) common to A will be A + p1 away. Eg. 20 has smaller prime factor as 2, so 20+2 is the next number that has a common factor with 20. Note 20+1 will not have any factor common with 21.

Let's get back to the problem. We were at finding common factors b/w z and 8z+1. Smallest prime factor of 8z is 2. Hence, 8z+1 will NOT have any common factor 8z. Since 2 is the smallest PRIME number in-general it can be said that 8z+1 and z will NOT have any common factor. Thus, x and y (above) will ONLY have 12 as GCD. QED!

PS: NOTE:
If we had,
y=12z
x=12 (2z+3)
Then statement (2) would have been insufficient.
Why?
z=1 => GCD=12
z=3 => GCD=36

However, with
y=12z
x=12 (2z+3)
Given z is not a multiple of 3,
It's sufficient.

You can try to imagine more such cases, and doing so should help you exercise your brain along the lines of factors b/w 2 numbers in general.

[tim]

Thanks for sharing your thoughts. Couldn't tell if there were any non-rhetorical questions in your post, but if there were please ask them again and we'll be glad to help..

[ccamankulor]

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

sequestered

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

y = 3m
u = 2k + 1 = 3, 5, 7, 9, 11

(3, 36) 3
(6, 60) 6
(9, 84) 3
(12, 108) 12
(15, 132) 3
(18, 156) 6
(21, 180) 3
(24, 204) 12

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

f(y) = 12z

f(x) = 96z + 12

Values of z greater than 12
are not factors of f(x), since
all factors of f(x) must be
factors of 96z and 12 simultaneously.

Thus, numbers greater than 12 are not common
factors of x and y, and the largest common
factor is known as 12.

[tim]

Thanks for sharing your thoughts. Couldn't tell if there were any non-rhetorical questions in your post, but if there were please ask them again and we'll be glad to help..

[ffearth]

RULE: If one number is b units away from another number, and b is a factor of both numbers, the greatest common factor of the two numbers is b.

Could you please give a proof of this rule? Thank you.

[tim]

i could, but proofs are irrelevant on the GMAT, so my time (and that of all the other instructors on the forum) is better spent answering questions that will help students on the GMAT. that said, i'd love to give you a proof if you can give me any reason to believe it'll help you on the GMAT.. :)

[ffearth]

I cannot memorize any rule if I don't understand why it applies. I believe that getting a very high score on the math part of the GMAT is only possible if one has a complete mastery and understanding of the subjects tested. Memorizing rules and formulas does not help mastering these concepts.
For this type of questions, a genuine understanding of number properties is the only way to excel.

If you give me the proof of this rule I will have a better understanding of number properties and I will maybe be able to deduce other rules that might be helpful on other hard problems.

I hope these reasons are enough for you to give me the proof.

Thank you.

[jnelson0612]

I cannot memorize any rule if I don't understand why it applies. I believe that getting a very high score on the math part of the GMAT is only possible if one has a complete mastery and understanding of the subjects tested. Memorizing rules and formulas does not help mastering these concepts.
For this type of questions, a genuine understanding of number properties is the only way to excel.

If you give me the proof of this rule I will have a better understanding of number properties and I will maybe be able to deduce other rules that might be helpful on other hard problems.

I hope these reasons are enough for you to give me the proof.

Thank you.

Hi there,
I'm sorry, but as Tim said, what you are asking for is really outside the scope of what we can do with this forum. I wish that we could help you, but we are swamped answering all the questions that we have.

One thing I will suggest is try to build your own understanding by plugging in different numbers that fit the concept. Play, explore, and see why this always holds true.

[ffearth]

that said, i'd love to give you a proof if you can give me any reason to believe it'll help you on the GMAT.. :)

I hope I gave good reasons and that you will give me a proof Tim :)

[tim]

RULE: If one number is b units away from another number, and b is a factor of both numbers, the greatest common factor of the two numbers is b.

Could you please give a proof of this rule? Thank you.

first off, everything Jamie said is right. but i wanted to write a proof for this one and was looking for you to give me an excuse. :)

let x = y - b, all integers. assume there is some integer a > b such that a is a factor of both x and y. then x = a*m and y = a*n for some integers m and n. now a*m = a*n-b or b = a(n-m). by this logic, a is a factor of b, which contradicts our assumption that a > b. thus there is no integer a > b such that a is a factor of both x and y.

but remember, Jamie was right: for the vast majority of test takers this sort of thing is way more than you need to succeed on the GMAT..

[ffearth]

RULE: If one number is b units away from another number, and b is a factor of both numbers, the greatest common factor of the two numbers is b.

Could you please give a proof of this rule? Thank you.

first off, everything Jamie said is right. but i wanted to write a proof for this one and was looking for you to give me an excuse. :)

let x = y - b, all integers. assume there is some integer a > b such that a is a factor of both x and y. then x = a*m and y = a*n for some integers m and n. now a*m = a*n-b or b = a(n-m). by this logic, a is a factor of b, which contradicts our assumption that a > b. thus there is no integer a > b such that a is a factor of both x and y.

but remember, Jamie was right: for the vast majority of test takers this sort of thing is way more than you need to succeed on the GMAT..

Many Thanks :D

[tim]

:)

[54siwei]

I encountered this question on the GMATPrep test. I am looking for a systematic way of solving this type of problems within 2 mins; Below is my approach. Could someone please verify it?


Statement1: x=12u, -->12 is a factor of x, u=all integers;
Statement2: y=12z, plug x=8y+12 and rewrite, --> x=12(8z+1), -->12 is a factor of x, (8z+1)=certain integers; so statement 2 is contained within statement 1 ; if statement1 is sufficient, statement 2 must be sufficient; kill a,c,d;
Between b and e:
12 is a common devisor of x and y, is 12 the GCF of x and y?
Yes if the GCF of z and8z+1 is 1;
plug in numbers for z and 8z+1
z=1, 8z+1=9;
z=2, 8z+1=17;
2=3,8z+1=25;
Z=9;8z+1=73;
Observed 1 is the GCF of z and8z+1.
Conclusion: statement2 is sufficient.

[tim]

first, if one statement is contained within the other, you cannot automatically eliminate D. second, plugging in a few values is not sufficient to conclude that something will always be the case..