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# Two positive integers a and b are divisible by 5, which is their

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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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EgmatQuantExpert
Two positive integers a and b are divisible by 5, which is their largest common factor. What is the value of a and b?

(1) The lowest number that has both integers a and b as its factors is the product of one of the integers and the greatest common divisor of the two integers.

(2) The smaller integer is divisible by 4 numbers and has the smallest odd prime number as its factor.

We will provide the OA in some time. Til then Happy Solving

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1) From task we know that GCD(a, b) = 5 and from statement we know that LCM equal to product of GCD and one of integers
so it can be such variant: a = 15 and b = 25; GCD(15, 25) = 5 and LCM(15, 25) = 75; GCD(15, 25) * 15 = 75
or it can be another variant: a = 10 and b = 25; GCD(10, 25) = 5 and LCM(10, 25) = 50; GCD(10, 25) * 10 = 50
Insufficient

2) From this statement we know that smaller integer divisible by for numbers and has 3 as a factor (smallest odd prime) and we know about factor 5
so we can make infer that this number equal to $$3^1* 5^1 = 15$$
But we know nothing about number b and this statement insufficient.

1 + 2) from second statement we know that a = 15 and from first statement we know that b = 25 Sufficient

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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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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 equations ensures a solution.

Two positive integers a and b are divisible by 5, which is their largest common factor. What is the value of a and b?

(1) The lowest number that has both integers a and b as its factors is the product of one of the integers and the greatest common divisor of the two integers.

(2) The smaller integer is divisible by 4 numbers and has the smallest odd prime number as its factor.

Normally, we use a=xG, b=yG (G=Greatest Common Factor, x and y are relative prime numbers: common factors is only 1) then L=xyG(L=Least Common Multiple).

In the original condition, a=5x, b=5y (x,y are relative prime numbers) therefore L=5xy. Since we have 4 variables (a,b,x,y) and 2 equations (a=5x, b=5y), we need 2 more equations to match the number of variables and equations and since there are 1 each in 1) and 2), the answer is likely C. Using 1) & 2) both we have L=5b=5xy, b=xy=5y thus x=5 and therefore a=5*5=25. Since b have smallest prime number as a factor, 3 is a factor of b and thus b=3*5=15. (The number of factors is 4 : 1,3,5,15).

But there are 2 cases: a=25, b=15 or a=15, b=25, therefore it is not unique and thus is not sufficient. Therefore the answer is E.
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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healthjunkie
Can you clarify in statement 1 where the second 5 comes from? I thought statement one basically says LCM= (a or b)*5 - does the second 5 come from the fact that a and b are also divisible by 5? Sorry for the confusing question!

Dear healthjunkie

You're right that St. 1 says LCM= (a or b)*5 . . . (1)

After this, we apply the property that (LCM of 2 numbers)*(GCD of 2 numbers) = Product of the two numbers themselves

So, in this case, we can write LCM*GCD = a*b . . .(2)

But, we are given that GCD = 5

So, substituting (1) and the value of GCD in (2), we get:

(a or b)*5*5 = a*b

Best Regards

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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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Can you clarify in statement 1 where the second 5 comes from? I thought statement one basically says LCM= (a or b)*5 - does the second 5 come from the fact that a and b are also divisible by 5? Sorry for the confusing question!
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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EgmatQuantExpert
Detailed Solution...

Yeah, I miss key pitfall in this task )

2. Familiarize yourself with all the names by which the test makers can call the GCD and the LCM.

I spend a lot of time trying to understand that all this convoluted phrases are just synonyms for GCD and LCM

Thanks for this task, really hard and interesting )
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
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smashbiker84
hi
though i got the correct answer, i would like a clarification on my line of thinking w.r.t statement 2 ( again :

it says the smallest integer is divisible by 4 numbers but not " only / exactly 4 numbers" in which case from Q STEM that 5 is their HCF and that 3 is also a factor , 15 would be the obvious inference..Am i wrong to infer that the number could also be a multiple of 15 ..?for example 30 , which is divisible by not only 4 but more than 4 numbers ( 8 to be precise) . also what is stated is that 3 is a factor of the smallest integer but what power of 3 is left to speculation, rite?

please let me know if i am wrong in my thought process . will help me attack such word problem ds questions with much more conviction.

regards

it says the smallest integer is divisible by 4 numbers but not " only / exactly 4 numbers"
Yes it's not stated clearly, but that was intended meaning: "divisible by 4 numbers only"

15 would be the obvious inference..Am i wrong to infer that the number could also be a multiple of 15 ..?for example 30
Yes, technically you are right.

I am not completely sure, but I think that if we see in question something like this: "number divisible by 4 numbers" or "number have 4 factors" this means exactly 4
and if we see "number divisible by more than 4 numbers" or "number have more than 4 factors" when we should apply logic that you use in your question.
IMHO
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
Hi,
Could you pls elaborate stmt 2 with the concept? I am not able to get that..

I could get that one of the factor is 3. and other is 5(from the question as a and b are divisible by 5). But stmt 2 states that smallest of a and b is disible by 4 numbers. we got 2 factors.. but how could we confirm the numbr to be 3*5=15 as other 2 factors are missing.. Or do we have a concept that (one of the factors of a)*HCF(a,b)=a? Please help... I am confusing a lot in this concept..
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
hi
though i got the correct answer, i would like a clarification on my line of thinking w.r.t statement 2 ( again :

it says the smallest integer is divisible by 4 numbers but not " only / exactly 4 numbers" in which case from Q STEM that 5 is their HCF and that 3 is also a factor , 15 would be the obvious inference..Am i wrong to infer that the number could also be a multiple of 15 ..?for example 30 , which is divisible by not only 4 but more than 4 numbers ( 8 to be precise) . also what is stated is that 3 is a factor of the smallest integer but what power of 3 is left to speculation, rite?

please let me know if i am wrong in my thought process . will help me attack such word problem ds questions with much more conviction.

regards
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
smashbiker84- It was rightly pointed out by Harley1980 when the question statement says divisible by 4 numbers, it surely means 4 numbers "only". If it's more than 4,that would be stated in the statements.

Regards
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
smashbiker84
hi
though i got the correct answer, i would like a clarification on my line of thinking w.r.t statement 2 ( again :

it says the smallest integer is divisible by 4 numbers but not " only / exactly 4 numbers" in which case from Q STEM that 5 is their HCF and that 3 is also a factor , 15 would be the obvious inference..Am i wrong to infer that the number could also be a multiple of 15 ..?for example 30 , which is divisible by not only 4 but more than 4 numbers ( 8 to be precise) . also what is stated is that 3 is a factor of the smallest integer but what power of 3 is left to speculation, rite?

please let me know if i am wrong in my thought process . will help me attack such word problem ds questions with much more conviction.

regards

I dont think you are wrong,,,, but the question did mention it is smallest of 2 integers and 30 is not less than 25 so 15 is the only valid option left Atleast that is what i understood
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
Hi,

I have another general question. If we are able to solve the question using just the question stem, irrespective of either statement 1 or statement 2, which answer choice should we be selecting in the data sufficiency question? Although rare, I found such a problem in the MGMAT official guide.

Thanks
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
himabindua
Hi,

I have another general question. If we are able to solve the question using just the question stem, irrespective of either statement 1 or statement 2, which answer choice should we be selecting in the data sufficiency question? Although rare, I found such a problem in the MGMAT official guide.

Thanks
Bindu

I have not seen any such question on the actual GMAT. If the original question stem itself is sufficient then the individual statements will not be needed and hence none of the options can be chosen. Such a question will defeat the purpose it being a DS question.

Do you mind posting the problem that you are talking about?
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
himabindua
Hi,

I have another general question. If we are able to solve the question using just the question stem, irrespective of either statement 1 or statement 2, which answer choice should we be selecting in the data sufficiency question? Although rare, I found such a problem in the MGMAT official guide.

Thanks
Bindu

Technically the answer would be D but GMAT won't give you such a question. There was 1 such GMAT Prep question though.
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
Two positive integers a and b are divisible by 5, which is their largest common factor. What is the value of a and b?

(1) The lowest number that has both integers a and b as its factors is the product of one of the integers and the greatest common divisor of the two integers.

(2) The smaller integer is divisible by 4 numbers and has the smallest odd prime number as its factor.

Sol: a = 5,10, 15 ... ; b= 5,10,15...
1. Lowest Integer with factors a & b can be written in the form of a*b*t where t can be +ve integer.
If we take a*b*t = a* 5 which gives us b*t = 5 since b min value of b is 5 and t is integer only value that satisfies is b=5
even we take a*b*t = b*5 => a =5
which means one number is 5 so the other number should be 10.
(a,b) can be (5,10) or (10,5) -> A is insuff
2. smaller is div by 4 and is also div by 3( small odd prime) hence smaller number div by 12
Possible values are 60, 120, ...
for 5 to be GCD the other number can be 65(>60), 125(>120).. No definite solution -> B insuff

Combing does not give any solution C is out.
Ans should E.
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
egmat Can you please clarify the statement 1 again and in a bit laymen method, I am confused how have you deduced the first statement and from where this extra 5 came, please use some example ?
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
Even if we combine both the statements , we will not be able to deduce whether a is greater than b or vice versa.The question specifically asks about values of a and b. Neither of the two stems help us decide a> b or a<b .

Hence, E.
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Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
Hi All,

basically this question does not require complicated math at all, even though I solved it using the math too.
But after some considerations I came up with this approach:

The questions asks us for the concrete values of a and b.
As long as the solution is not a=b (which is explicitely denied in statement 2 and GMAT statements never contradict each other) we can either find at least 2 values or the information is not sufficient at all.
As neither in statement 1 nor in statement 2 nor in the question stem itself is given any clue, which of the value has to be assigned to a and which to b, the answer choice will always be E.

The situation would change, if e.g. statement 2 were "The smaller number a ..." and you would have to do the math to come up with C instead of E.

But the question is, is it worthwhile to watch out for such a 'simple' solution or not (i.e. hurl oneself immediately into the math).
I guess such 'simple' cases are quite rare and it definitely costs time to check for such cases.
It will certainly cost you ~ 10 secs to check for the above situation and for a positive will shortcut the answer so that you save 100 secs (it took me 180 seconds to solve this one),
so with 15 DS questions, it will cost you ~ 150 secs for the check and, as there is probably only one 'simple' case, will save you ~ 100 secs,
the bargain is negative.