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If a and b are positive integers divisible by 6, is 6 the [#permalink]
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02 Sep 2010, 23:28
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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 What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition?
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Re: If a and b are positive integers [#permalink]
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note that a=3b does not mean that 6 will be the gcd. example 6, 18 is 6, but 12, 36 is 12 not suff for a =2b+6= 2(b+3); take b=6,a=18, gcd =6 ; take b=12,a=30; gcd=6; take b=18,a=42; gcd=6 suff for an algebraic proof: a=6r, b=6s 2. a=3b means 6r=18s or r=3s; a=6s b=18s; cannot conclude about gcd a=2b+6 means r=2s+1; a=6(2s+1); b=6s no common factors; you can conclude that 6 will be gcd.
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Re: If a and b are positive integers [#permalink]
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gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b?Given: \(a=6x\) and \(b=6y\). Question: is \(GCD(a,b)=6\)? Now, If \(x\) and \(y\) share any common factor >1then \(GCD(a,b)\) will be more than 6 if not then \(GCD(a,b)\) will be 6. (1) \(a=2b+6\) > \(6x=2*6y+6\) > \(x=2y+1\) > \(x\) and \(y\) do not share any factor >1, as if they were we would be able to factor out if from \(2y+1\). Sufficient. (2) \(a=3b\) > clearly insufficient. Answer: A. There is also a general rule: 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 \(b\).For example if \(a\) and \(b\) are multiples of 7 and \(a=b+7\) then 7 is GCD of \(a\) and \(b\). So if we apply this rule to (1) \(a=2b+6\) > both \(a\) and \(2b\) are multiples of 6 and are 6 apart, so GCD of \(a\) and \(2b\) is 6, hence GCD of \(a\) and \(b\) is also 6. Sufficient. Hope it helps.
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Re: If a and b are positive integers [#permalink]
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03 Sep 2010, 11:57
Hey Bunuel....thanx for the rule stated...
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Re: If a and b are positive integers [#permalink]
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03 Sep 2010, 12:04
My appreciation to you guys for taking your time to provide this informative response; truly grateful. @Bunuel, x and y do not share any further factors ( x=2y+1  consecutives), GCD(a,b) = 6. Also, thanks for the distance apart info. Regards to both of you Bunuel and Mainhoon.
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Re: If a and b are positive integers [#permalink]
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21 Sep 2010, 07:05
Thanks for the solution
Bunuel, may be you should post a collection of all such rules.



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Re: If a and b are positive integers [#permalink]
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17 Feb 2011, 00:26
Bunuel wrote: gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b? Given: \(a=6x\) and \(b=6y\). Question: is \(GCD(a,b)=6\)? Now, If \(x\) and \(y\) share any common factor >1then \(GCD(a,b)\) will be more than 6 if not then \(GCD(a,b)\) will be 6. (1) \(a=2b+6\) > \(6x=2*6y+6\) > \(x=2y+1\) > \(x\) and \(y\) do not share any factor >1, as if they were we would be able to factor out if from \(2y+1\). Sufficient. (2) \(a=3b\) > clearly insufficient. Answer: A. There is also a general rule: 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 \(b\).For example if \(a\) and \(b\) are multiples of 7 and \(a=b+7\) then 7 is GCD of \(a\) and \(b\). So if we apply this rule to (1) \(a=2b+6\) > both \(a\) and \(2b\) are multiples of 6 and are 6 apart, so GCD of \(a\) and \(2b\) is 6, hence GCD of \(a\) and \(b\) is also 6. Sufficient. Hope it helps. Thanks for the rule. I picked A as I knew there is "some" rule for common multiples a an integer with that integer as the difference between the common multiples but was not recollecting it..
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Re: If a and b are positive integers [#permalink]
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17 Feb 2011, 00:32
144144 wrote: these considered to be hard questions? Definitely above 600 level.
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Re: If a and b are positive integers [#permalink]
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17 Feb 2011, 18:33
Very good approach to alternative 1. Bunuel wrote: gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b? Given: \(a=6x\) and \(b=6y\). Question: is \(GCD(a,b)=6\)? Now, If \(x\) and \(y\) share any common factor >1then \(GCD(a,b)\) will be more than 6 if not then \(GCD(a,b)\) will be 6. (1) \(a=2b+6\) > \(6x=2*6y+6\) > \(x=2y+1\) > \(x\) and \(y\) do not share any factor >1, as if they were we would be able to factor out if from \(2y+1\). Sufficient. (2) \(a=3b\) > clearly insufficient. Answer: A. There is also a general rule: 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 \(b\).For example if \(a\) and \(b\) are multiples of 7 and \(a=b+7\) then 7 is GCD of \(a\) and \(b\). So if we apply this rule to (1) \(a=2b+6\) > both \(a\) and \(2b\) are multiples of 6 and are 6 apart, so GCD of \(a\) and \(2b\) is 6, hence GCD of \(a\) and \(b\) is also 6. Sufficient. Hope it helps.



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Re: If a and b are positive integers [#permalink]
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05 Feb 2012, 02:43
Bunuel wrote: gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? If a and b are positive integers divisible by 6, is 6 the greatest common divisor of a and b? Given: \(a=6x\) and \(b=6y\). Question: is \(GCD(a,b)=6\)? Now, If \(x\) and \(y\) share any common factor >1then \(GCD(a,b)\) will be more than 6 if not then \(GCD(a,b)\) will be 6. (1) \(a=2b+6\) > \(6x=2*6y+6\) > \(x=2y+1\) > \(x\) and \(y\) do not share any factor >1, as if they were we would be able to factor out if from \(2y+1\). Sufficient. (2) \(a=3b\) > clearly insufficient. Answer: A. There is also a general rule: 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 \(b\).For example if \(a\) and \(b\) are multiples of 7 and \(a=b+7\) then 7 is GCD of \(a\) and \(b\). So if we apply this rule to (1) \(a=2b+6\) > both \(a\) and \(2b\) are multiples of 6 and are 6 apart, so GCD of \(a\) and \(2b\) is 6, hence GCD of \(a\) and \(b\) is also 6. Sufficient. Hope it helps. But if a and b are both divisible of 6, means that both are even, therefore at least both of them should be divisible by 2.... I am right????? I do not understand why (1) is valid... thanks!!



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Re: If a and b are positive integers [#permalink]
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Saurajm wrote: But if a and b are both divisible of 6, means that both are even, therefore at least both of them should be divisible by 2.... I am right????? I do not understand why (1) is valid... thanks!! Yes, both are divisible by 6, which means that they are divisible by 2 and 3. Next, we have that \(a=6x\) and \(b=6y\). Consider two cases: 1. \(x\) and \(y\) share some common factor >1: for example \(x=2\) and \(y=4\) then \(a=12\) and \(b=24\) > \(GCD(a,b)=12>6\); 2. \(x\) and \(y\) DO NOT share any common factor >1: for example \(x=5\) and \(y=2\) then \(a=30\) and \(b=12\) > \(GCD(a,b)=6\). From (1) we have that > \(x=2y+1\) > \(x\) is one more than multiple of \(y\). For example: \(x=3\) and \(y=1\) OR \(x=5\) and \(y=2\) OR \(x=7\) and \(y=3\) ... as you can see in all these cases x and y do not share any common factor more than 1. Now, as we concluded above if \(x\) and \(y\) DO NOT share any common factor >1, then \(GCD(a,b)=6\). Or another way: \(b=6y\) and \(a=6(2y+1)\). \(2y\) and \(2y+1\) are consecutive integers and consecutive integers do not share any common factor 1. As \(2y\) has all factors of \(y\) then \(y\) and \(2y+1\) also do not share any common factor but 1, which means that 6 must GCD of \(a\) and \(b\) Similar questions to practice: whatisthegreatestcommonfactorofxandy1xandyare109273.htmlxandyarepositiveintegerssuchthatx8y12whatisthe126743.htmlHope it helps.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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22 Feb 2012, 09:25
gmatbull wrote: 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
I suppose my doubt was not conveyed properly. By contradiction I meant for a value of b, we get two different a(s) by two statements. Or else, (18,6) is the only solution possible.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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14 Mar 2012, 04:15
Hi Bunuel, Can you explain the following rule with few examples.
Given: and . Question: is ? Now, If x and y share any common factor >1then GCD (a,b) will be more than 6 if not then GCD (a,b) will be 6.



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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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pavanpuneet wrote: Hi Bunuel, Can you explain the following rule with few examples.
Given: and . Question: is ? Now, If x and y share any common factor >1then GCD (a,b) will be more than 6 if not then GCD (a,b) will be 6. Sure. Both \(a\) and \(b\) are multiples of 6 > \(a=6x\) and \(b=6y\). Consider two cases: A. \(x\) and \(y\) do not share any common factor >1, for example \(a=6*2=12\) and \(b=6*3=18\) > GCD(a,b)=6. As you can see 2 and 3 did not contribute any common factor to the GCD; B. \(x\) and \(y\) share some common factor >1, for example \(a=6*2=12\) and \(b=6*4=24\) > GCD(a,b)=12, here 2 and 4 contributes common factor 2 to the GCD. Hope it's clear.
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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16 Mar 2012, 01:43
Thank you once again for the prompt response. It's clear.



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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? Given a and b are positive integers divisible by 6 From st 1, we have a = 2b+ 6 b is of the form b= 6c where c is a positive integer therefore we have a = 2*6c+6 > a= 6 (2c+1) b= 6c Now HCF of a and b will be 6 as a= 2*3*(2c+1) b=2*3 Only common factor is 6 So, Ans is A alone sufficient. We can remove options B, C and E as possible answers Not st 2 says a= 3b taking b = 6c we have a=18c a= 2*3^2*c b=2*3 c HCF will be 2*3*c that is 6c Now if c=1 then 6 is a the HCF of both a and b but if c=2 then 12 is HCF Since 2 ans choices are possible. Ans will be A to this Question
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? Question: 6 * x =a and 6 * y =b we need to find if 6 is the GCD? YES or NO question. so, basically if we can find a single common factor in x and y, thats it its not a GCD, or if we cant find one then that should also work for us. (1) a = 2b + 6 6x=12y + 6 x = 2y +1 =>No matter what you do, this will always result in an no common factor. Thus 6 is the only GCD =>Sufficient (2) a = 3b 6x = 18y x=3y Take y=8, and Y=2 =>This is clearly Not sufficient. Ans: A
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Re: If a and b are positive integers divisible by 6, is 6 the [#permalink]
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30 Dec 2013, 10:28
gmatbull wrote: 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
What is the fastest (perhaps, algebra?) means of solving the question besides random plugging numbers under test condition? I think that the best approach is a combination of both Check this out We are told that a = 6k , b = 6m Is gcf > 6? Statement 1 A = 2b + 6 Replacing we get 6k = 12m + 6 Now if m = 1, k = 3 m = 2, k = 5 As you see there will not be any integers m,k with GCF>1 Hence sufficient Statement 2 a=3b 6k=18m k=6m When m=1, k=6 When m=2, k=12 BINGO, m and k share common factor 2 So replacing we get that 12 is also a factor of both 'a' and 'b' Then we have to possible answers and thus the statement is not sufficient Answer is A Hope it helps Cheers! J




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