NEW HERE? LEARN MORE
closeclose
Last visit was: 26 Apr 2024, 18:58 It is currently 26 Apr 2024, 18:58
Decision Tracker
My Rewards
New posts
New comers' posts

Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel

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

SORT BY:
Date
Kudos
   1   2 
Tags:
Find Similar Topics 
Show Tags
Hide Tags
avatar
pranav18
Intern
Intern
Joined: 13 Mar 2016
Posts: 1
Own Kudos [?]: [0]
Given Kudos: 34
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   19 Apr 2017, 10:06
I just have one doubt wrt statement 1. Where is that another 5 coming from?
avatar
B
Sateni7628
Intern
Intern
Joined: 18 Jan 2018
Posts: 48
Own Kudos [?]: 30 [0]
Given Kudos: 34
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   18 Feb 2018, 03:00
Tough one to interpret & crack correctly.
User avatar
G
testcracker
Manager
Manager
Joined: 24 Mar 2015
Status:love the club...
Posts: 220
Own Kudos [?]: 112 [0]
Given Kudos: 527
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   30 Mar 2018, 05:55
EgmatQuantExpert wrote:
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


Harsh
hi

since both integers have 5 as their prime factors, and statement 2 says that smaller integer has 3 as its prime factor, we can say one integer is 5 * 3 = 15

now, can you please tell me why to assume that the smaller integer doesn't have any other prime factors other than 3 and 5..?

is this because, 15 has exactly 4 factors, a condition met ?

thanks in advance
User avatar
P
LevanKhukhunashvili
Senior Manager
Senior Manager
Joined: 13 Feb 2018
Posts: 386
Own Kudos [?]: 385 [0]
Given Kudos: 50
GMAT 1: 640 Q48 V28
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   15 Oct 2018, 00:48
I arrived at 15 and 25 and then was tempted to smash C omitting that I still didn't know which one is "a" and which one is "b"

Sad story for Q50-Q51 hunters :)
User avatar
P
fskilnik
Tutor
Joined: 12 Oct 2010
Status:GMATH founder
Posts: 893
Own Kudos [?]: 1356 [0]
Given Kudos: 56
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   15 Oct 2018, 06:06
Expert Reply
Quote:
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 POSITIVE 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 EXACTLY 4 POSITIVE numbers and has the smallest odd prime number as ONE OF its factors.

\(\left. \matrix{\\
a,b\,\, \ge 1\,\,{\rm{ints}} \hfill \cr \\
{\rm{GCD}}\left( {a,b} \right) = 5\,\, \hfill \cr} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{\\
\,a = 5M\,\,,\,\,M \ge 1\,\,{\mathop{\rm int}} \, \hfill \cr \\
\,b = 5N\,\,,\,\,N \ge 1\,\,{\mathop{\rm int}} \, \hfill \cr} \right.\,\,\,\,\,\,M,N\,\,{\rm{relatively}}\,\,{\rm{prime}}\)

\({\rm{?}}\,\,\,{\rm{:}}\,\,\,{\rm{a,}}\,{\rm{b}}\)

Let´s go straight to (1+2), because it is easy to BIFURCATE (1) and (2) together (and this guarantees that each alone is also insufficient):

\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5 \cdot 3,5 \cdot 5} \right)\,\,\,\,\left( * \right)\, \hfill \cr \\
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {5 \cdot 5,5 \cdot 3} \right)\,\,\,\,\left( {**} \right)\,\, \hfill \cr} \right.\)

\(\left( * \right)\,\,\,\left\{ \matrix{\\
LCM\left( {a,b} \right) = 3 \cdot {5^2} = \left( {5 \cdot 3} \right) \cdot 5 = a \cdot GCD\left( {a,b} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\left( 1 \right)\,\,{\rm{satisfied}} \hfill \cr \\
\,{\rm{5}} \cdot {\rm{3}}\,\,\,{\rm{has}}\,\,{\rm{exactly}}\,\,{\rm{4}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{and}}\,\,{\rm{3}}\,\,{\rm{is}}\,\,{\rm{one}}\,\,{\rm{of}}\,\,{\rm{its}}\,\,{\rm{factors}}\,\,\,\,\,\, \Rightarrow \,\,\,\left( 2 \right)\,\,{\rm{satisfied}} \hfill \cr} \right.\)

\(\left( {**} \right)\,\,\,{\rm{analogous}}\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
User avatar
P
Probus
Manager
Manager
Joined: 10 Apr 2018
Posts: 187
Own Kudos [?]: 448 [0]
Given Kudos: 115
Location: United States (NC)
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   15 Apr 2019, 14:06
MathRevolution wrote:
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.




Hi Here are my two cents for this question,

If P and Q are two positive integers, then we can write

P = AL and Q=AM where A is the HCF(P,Q) and L and M are co primes

LCM(PQ)= A*L*M
HCF(PQ)=A


Here we are told that a and b are integers and their HCF is 5,
then a= 5X and B=5Y where X, Y are co primes.

Statement 1 tells us that LCM (a,b)= 5 * a or LCM (a,b)= 5 * b
We know that LCM(a,b)=5XY

so either 5XY= 5*5*X or 5XY= 5*5*Y
which means wither X is 5 or Y is 5 so we can say either a= 25 or b=25

From Statement (II) we have
Purpose of statement 2 is tell us that ( Power of other prime factor is 1 ) one of the integers has total 4 factors, and one of the factors is smallest Odd Prime.
Smallest odd Prime is 3 and its power is 1 = \(3^{1}\)
We know Either if a is small and 5 is one of factors then 3 is other factor .
So if we say a is small of the two then a = 15 , if we consider b as small of two we have b =15

Combining Both we have
Either a=25 then b=15
or a=15 and b=25

But conclusively cannot point any particular value if a and b
User avatar
L
Fdambro294
VP
VP
Joined: 10 Jul 2019
Posts: 1392
Own Kudos [?]: 542 [0]
Given Kudos: 1656
Send PM
Re: Two positive integers a and b are divisible by 5, which is their [#permalink] New post   06 Jun 2023, 08:56
EgmatQuantExpert wrote:
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.

This is

Ques 4 of The E-GMAT Number Properties Knockout



Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts :)




So basically, just realize that neither statement provides enough information that we can use to assign the exact value to variable “a” or variable “b”, even if it is possible to find the unordered solution.

click E without performing any calculations.

Great question with respect to the concepts tested, however.

Posted from my mobile device
GMAT Club Bot
gmatclubot
Re: Two positive integers a and b are divisible by 5, which is their [#permalink]
06 Jun 2023, 08:56
   1   2 
Moderator:
Bunuel
Math Expert
92948 posts

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions