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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]
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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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]
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]
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

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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.

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