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Re: Number Properties - Divisibility and Primes [#permalink]

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13 Feb 2011, 10:10

maryann wrote:

What is the greatest common factor of x and y

1. x and y are both divisible by 4 2. x - y = 4

1. is not sufficient, x and y can be 16 and 32, in which case the GCF is 16, or 4 and 8, in which case the GCF is 4

2. gives even less information, x and y can be anything as long as their difference is 4, so we have no info about either of their factors, e.g., x=19, y=15 or x=4, y=8.

combined: you can write y = 4n, with n being any integer, since y is divisible by 4. Then x = 4n + 4 = 4(n+1). Now, n and n+1 don't have a common factor greater than 1(*), and therefore the GCF = 4. Answer is C.

(*)You don't need to prove this for the answer, but it's always true and follows from the fact that if n is even, n+1 is odd (or vice versa). If we assume they have a GCF>1, it has to be odd, since an odd number cannot have an even factor.

lets say n is even, so we can write as

n = q * w (where q is the GCF with n+1, which must be odd, and w is the product of all other factors, which must be even)

n + 1 = q * r (where r is the product of all of n+1's other factors, and both q and r are odd)

if you replace q in the second equation, it can be written as

n * (w - 1) = r - w ( Even * Odd ) = Odd - Even => Even = Odd, which is a contradiction.

Last edited by cmv on 13 Feb 2011, 10:21, edited 1 time in total.

Re: Number Properties - Divisibility and Primes [#permalink]

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13 Feb 2011, 10:31

Hi cmv,

Thanks for the solution. but i am still a bit confused with y = 4n and x = 4n + 2 = 4(n+1). (how did you get this equations from the statements) I dont understand how n and n+1 dont have a common factor greater than 1(*).

Re: Number Properties - Divisibility and Primes [#permalink]

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13 Feb 2011, 10:34

1

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If you take two consecutive numbers, one will be odd and the other will be even... thus, they will not have any common factors between them! Thus they're co-prime...

and C is when you can answer the question using both the options given but not either of them alone!
_________________

"Wherever you go, go with all your heart" - Confucius

Re: Number Properties - Divisibility and Primes [#permalink]

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13 Feb 2011, 10:50

1

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maryann wrote:

Hi cmv,

Thanks for the solution. but i am still a bit confused with y = 4n and x = 4n + 2 = 4(n+1). (how did you get this equations from the statements) I dont understand how n and n+1 dont have a common factor greater than 1(*).

how has the above deduction let to the answer C?.

Thanks Maryann

1. says that they are both multiples of 4, therefore we can write either of them as 4 * n, where n is any integer

2. says that the difference between them is 4, which, when you combine with the info from 1., means that they are *consecutive* multiples of 4, like 12 and 16, or 20 and 24. So

y = 4n (from 1.) => x = y + 4 (from 2.) => x = 4n + 4 => x = 4(n+1)

so, now you have x factorized into "4" and "n+1", and y factorized into "4" and "n". Either their GCF is 4, or their GCF is the GCF of n and n+1. If you know that n and n+1 don't have a GCF greater than one, then the GCF for x and y must be 4, and that's why the answer is C.

n and n+1 are two consecutive integers, one even, and one odd. The fact that two consecutive integers don't have a common factor greater than 1 is a bit trickier, but probably something that is good to remember for the GMAT. In the original answer I proved this by assuming that they do have a GCF, and then reaching a contradiction, if you don't understand how that was done let me know and I'll try to break it down further.

Stmnt 1: We know now that 4 is a factor of both. But is it the highest common factor, we do not know yet. There could be another factor common between x and y and hence highest common factor could be greater than 4. e.g. 4 and 16 have 4 as highest common factor but 12 and 36 have 12 as the highest common factor though both pairs have 4 as a common factor. Stmnt 2: We know that x and y differ by 4. So they could have any of 1/2/4 as their highest common factor (Explanation given below) e.g. 7 and 11 have 1 as common factor while 2 and 6 have 2 as greatest common factor.

Taking both together: From stmnt 1, x and y have 4 as a common factor. From stmnt 2, x and y have one of 1/2/4 as highest common factor. Hence 4 is the highest common factor.

Answer (C).

Explanation:

Notice a few things about integers: -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16......

Every number is a multiple of 1 Every second number is a multiple of 2 Every third number is a multiple of 3 Every fourth number is a multiple of 4 and so on...

If I pick any 2 consecutive integers, one and only one of them will be a multiple of 2: e.g. I pick 4, 5 (4 is a multiple of 2) or I pick 11, 12 (12 is a multiple of 2) etc..

If I pick any 3 consecutive integers, one and only one of them will be a multiple of 3: e.g. I pick 4, 5, 6 (6 is a multiple of 3) or I pick 11, 12, 13 (12 is a multiple of 3) etc..

This means that if I pick any two consecutive integers, they will have no common factor other than 1. (Say if 5 was their common factor, the numbers would be at least 5 apart e.g. 5 and 10.They cannot be consecutive. If 11 was their common factor, the numbers would be at least 11 apart e.g. 11 and 22. They cannot be consecutive. etc)

If I pick two integers with a difference of 4 between them, the only common factors (other than 1) they can have are 2 and/or 4 e.g. 2 and 6 have 2 as a common factor. 4 and 8 have 2 and 4 as common factors.
_________________

1. Two consecutive integers are co-prime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1.

2. For two distinct positive integers \(a\) and \(b\) (\(a>b\)): \(GCD(a,b)\leq{a-b}\), greatest common divisor of two distinct positive integers cannot be more that their positive difference. Proof: any common factor of two integers is also a factor of their sum and difference, hence GCD of two distinct integers cannot be more than the positive difference between them as in this case GCD must also be a factor of a difference which is less then it, which is impossible.

For example if \(a=15\) and \(b=10\) then gretest common divisor of 15 and 10 cannot be more than 15-10=5 (as number more than 5 cannot be divisor of 5, which is the difference between 15 and 10).

3. 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 \(b\). This is a derivation from the property above: if \(k\) is a factor of both \(a\) and \(b\) and \(a-b=k>0\) then from above \(GCD(a,b)\leq{k}\) but as we know that \(k\) is a divisor of both \(a\) and \(b\) then \(GCD(a,b)={k}\).

For example if \(a\) and \(b\) are multiples of 7 and \(a=b+7\) then 7 is GCD of \(a\) and \(b\).

BACK TO THE ORIGINAL QUESTION:

What is the greatest common factor of x and y ?

(1) x and y are both divisible by 4 --> clearly insufficient: of course 4 itself could be the GCD but it's also possible that x and y share some other common factor more than 4, for example 5 (in this case both will be divisible by 20), or 8, or 4,000,000 ... so GCD could be more than 4 as well. Though from this statement we know that GCD cannot possibly be less than 4.

(2) x - y = 4 --> according to the property #2: \(GCD(x,y)\leq{4}\), but still insufficient.

(1)+(2) From (1) \(GCD\geq{4}\) and from (2) \(GCD\leq{4}\) --> \(GCD=4\). Sufficient.

Re: What is the greatest common factor of x and y 1. x and y are [#permalink]

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26 Oct 2012, 11:35

S1 is insufficient since the numbers could be any multiple of 4 and can have common factors greater and other than 4 and 1. S2 is also insufficient. It could just be 65-61 = 4 Combining S1 & S2 - Consecutive multiples of 4. So the highest common factor is 4.
_________________

I will rather do nothing than be busy doing nothing - Zen saying

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the greatest common factor of x and y ?

(1) x and y are both divisible by 4 (2) x - y = 4

In the original condition, there are 2 variables(x,y), which should match with the number of equations. So you need 2 equations as well. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. In 1) & 2), x=4n, y=4m -> x-y=4n-4m=4, n-m=1 and greatest common factor(GCD) of n and m can only be 1. Then, GCD(x,y)=4, which is unique and therefore sufficient. So, the answer is C. In case of 1) and 2) respectively, it is not unique and therefore not sufficient. So, the answer is C.

-> For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
_________________

Re: What is the greatest common factor of x and y ? [#permalink]

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16 Mar 2016, 09:46

We just need to remember => If a and b are integers and both are multiples of C and are als C units apart from each other then C is the GCD of a and b hence C
_________________

Re: What is the greatest common factor of x and y ? [#permalink]

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22 Apr 2016, 09:54

A lot of solutions on this page already. Here is mine Clearly both 1 and 2 are alone insufficient. combining them => x=4a y=4b and a=b+1 where b and b+1 will always be coprimes. Hence We can say here that the GCD will be 4 Smash C
_________________

What is the greatest common factor of x and y ? [#permalink]

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11 Jul 2016, 09:19

Statement 1) Insufficient Case 1) x=12 (2*2*3) y=16 (2*2*2*2) ==> GCF=4 Case 2) x =12 (2*2*3) y=24 (2*2*2*3) ===> GCF = 6 OPTION A AND D out

Statement 2) Insufficient Case 1) x=15 , y= 11 ==> 15-11=4 ==>GCF =1 Case 1) x=16 , y= 12 ==> 16-12=4 ==>GCF =4 OPTION B out

Merge both statements x and y has to be the multiple of 4 and their difference have to be 4 Therefore X and Y are two consecutive multiple of 4 For example :- 4 & 8 or 12 & 16 or 24 & 28 ... Now if one of them is the form of= 4*even Number , the other will be= 4*odd number (because they are consecutive multiples of 4*(1,2,3,4,5,6,7,8,9......)) Therefore these two will have only 4 and 1 as the common factor (an odd and even number share no common multiple except 1) Therefore GCF=4*1= 4

SUFFFICIENT

ANSWER IS C

maryann wrote:

What is the greatest common factor of x and y ?

(1) x and y are both divisible by 4 (2) x - y = 4

_________________

Posting an answer without an explanation is "GOD COMPLEX". The world doesn't need any more gods. Please explain you answers properly. FINAL GOODBYE :- 17th SEPTEMBER 2016. .. 16 March 2017 - I am back but for all purposes please consider me semi-retired.

Hi, why can't i see the choices? Please help. Thanks

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked. D. EACH statement ALONE is sufficient to answer the question asked. E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Re: What is the greatest common factor of x and y ? [#permalink]

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10 Sep 2017, 02:43

Bunuel wrote:

lheiannie07 wrote:

Hi, why can't i see the choices? Please help. Thanks

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked. B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked. C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked. D. EACH statement ALONE is sufficient to answer the question asked. E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Hope this helps.

Thank you so much. I am new here so i need some guidance. Thanks a lot

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