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

Question

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 :lol:

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EgmatQuantExpert · Accepted Answer

Detailed Solution

Step-I: Given Info

We are given two positive integers a and b such that their highest common factor is 5. We are asked to find the values of a and b.

Step-II: Interpreting the Question Statement

Since, the highest common factor of a and b is 5, it implies that the GCD of (a, b) = 5. So, we can deduce that both a and b definitely have 5 as one of their prime factors. Let’s use the information given in the statements to see if we can find out the values of a and b.

Step-III: Statement-I

Statement-I tells us that the LCM (a, b) is the product of one of the integers of a and b with GCD (a, b) i.e. 5. Since, we know that the product of the numbers is equal to the product of their LCM & GCD, we can write
LCM * GCD = a * b

⇨ 5 * (a or b) * 5 = a * b
Simplifying this statement would give us the value of one of the number as 25. Since, we do not have any information about the value of the second number, statement-I is insufficient.

Step-IV: Statement-II

Statement-II tells us that the smaller of the two integers a and b is divisible by 4 factors and has the smallest odd prime number (i.e. 3) as its factor. So, the integer can be represented as:
x * y or x^3.
Since, we know that both the integers have 5 as one of its prime factors (since 5 is the GCD) so, we can represent the integer as product of two prime factors i.e. x * y = 3 * 5= 15.
Now, we know that one of the integers has the value of 15, but we are not given any information about the value of the second integer.

Hence, Statement-II is insufficient to answer the question.

Step-V: Combining Statements I & II

Statement- I tells us that one of the integers of a and b has the value as 25. Statement-II tells us that the other integer has the value as 15. This would mean that:
Either a= 25 and b =15 or
a = 15 and b =25.
Hence, we can’t say with certainty the specific values of a and b.

Thus, combining St-I & II is also not sufficient to answer the question.
Hence, the answer is Option E.

Key Takeaways

1. The prime factors of the GCD of a set of numbers would also be the prime factors of the numbers themselves.
2. Familiarize yourself with all the names by which the test makers can call the GCD and the LCM.

For example,
• GCD is also known as the HCF
• GCD can also be described as ‘the largest number which divides all the numbers of a set’
• LCM of a set of numbers can also be described as ‘the lowest number that has all the numbers of that set as it factors’

3. Read the question carefully

Harley1980 I hope you realized where did you miss

Regards
Harsh

Regards Harsh https://e-gmat.com/blogs/wp-content/uploads/2018/08/Quant-workshop_Webinar-e1533628475738.jpg

EgmatQuantExpert · Answer

Hi  sheolokesh - Let's assume a number X= P1^a * P2^b, where P1 & P2 are its prime factors. The total number of factors of X would be represented as (a+1)*(b+1) where a, b are the no. of times a prime factor is repeated in the prime factorization of a number. Let me explain you with an example:

Assume a number 12, now 12 can be written as 12 = 2^2 * 3^1. Hence, in this case, the total number of factors of 12 would be (2+1)* (1+1) = 6.
Hence 12 would have a total of 6 factors namely {1, 2,3,4,6 and 12}.

So, in the question when we are saying that the smaller number has 4 factors, it would mean that number can be expressed as either P1^3 or P1* P2 where P1,P2 are its prime factors. Since, we know that the number has 2 prime factors, therefore the number would be expressed as P1 * P2, in this case 3* 5 = 15.

Hope this is clear!

Regards
Harsh

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Harley1980 · Answer

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

Answer is C

Harley1980 · Answer

Yeah, I miss key pitfall in this task ) And yeah, you are absolutely right about this 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 )

sheolokesh · Answer

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..

smashbiker84 · Answer

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

Harley1980 · Answer

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

EgmatQuantExpert · Answer

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
Harsh

healthjunkie · Answer

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!

EgmatQuantExpert · Answer

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 Hope this clarified your doubt

Best Regards Japinder

RamyaM · Answer

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

himabindua · Answer

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

ENGRTOMBA2018 · Answer

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?

Bunuel · Answer

Technically the answer would be D but GMAT won't give you such a question. There was 1 such GMAT Prep question though.

PerptualKnite56 · Answer

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.

shekharshrek · Answer

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 ?

samusa · Answer

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.

Hence, E.

MathRevolution · Answer

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.

ringogutzeit · Answer

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.

What is your opinion or your experience?
Are there other 'simple' cases that might make it worthwhile to watch out for them in a bunch?
Is there some statistics available to confirm either way - check first or hurl into math immediately?

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

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

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