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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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New post 19 Apr 2017, 10:06
I just have one doubt wrt statement 1. Where is that another 5 coming from?
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Re: Two positive integers a and b are divisible by 5, which is their  [#permalink]

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New post 18 Feb 2018, 03:00
Tough one to interpret & crack correctly.
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Re: Two positive integers a and b are divisible by 5, which is their  [#permalink]

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

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

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New post 15 Oct 2018, 06:06
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.
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Re: Two positive integers a and b are divisible by 5, which is their  [#permalink]

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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
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Re: Two positive integers a and b are divisible by 5, which is their   [#permalink] 15 Apr 2019, 14:06

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