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Quotient of positive integers r and s

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Quotient of positive integers r and s

by gander123 » Sun Jan 13, 2013 4:11 am
Hey,

QDS03324 GMAT Prep Question Pack 1:

"If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r.

(2) Every prime factor of s is also a prime factor of r."

OA: A

OA Explanation for (1):
The integer s is by definition a factor of itself. From this, every factor of s is also a factor of r. Therefore r/s, must be an integer; SUFFICIENT.
For the sake of shortness, I leave the official answer for statement (2) out here.

My Questions:

1. To evaluate statement (1) I initially chose r = 6 (containing factors 2 and 3) and s = 18 containing factors 2 and 3, 3). In this case. 6/18 = 1/3 which certainly is no integer. The only plausible explanation for the wrongness of my numbers is that I did not choose two integers with the same
number of factors.

2. So does the phrasing
every factor
here mean "the same number/amount of" factors?


I would repeat this mistake thousands of times, since it really isnt quite obvious that statement (1) meant the same amount of factors..

What would you do ?


cheers,

Tobi
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by GMATGuruNY » Sun Jan 13, 2013 4:53 am
gander123 wrote:Hey,

QDS03324 GMAT Prep Question Pack 1:

"If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r.

(2) Every prime factor of s is also a prime factor of r."

OA: A

OA Explanation for (1):
The integer s is by definition a factor of itself. From this, every factor of s is also a factor of r. Therefore r/s, must be an integer; SUFFICIENT.
For the sake of shortness, I leave the official answer for statement (2) out here.

My Questions:

1. To evaluate statement (1) I initially chose r = 6 (containing factors 2 and 3) and s = 18 containing factors 2 and 3, 3). In this case. 6/18 = 1/3 which certainly is no integer. The only plausible explanation for the wrongness of my numbers is that I did not choose two integers with the same
number of factors.

2. So does the phrasing
every factor
here mean "the same number/amount of" factors?


I would repeat this mistake thousands of times, since it really isnt quite obvious that statement (1) meant the same amount of factors..

What would you do ?


cheers,

Tobi
In statement 1, r=6 and s=18 do not satisfy the condition that EVERY factor of s is also a factor of r.
For example, 9 and 18 are factors of s=18 but are NOT factors of r=6.

Statement 1: Every factor of s is also a factor of r.
For EVERY factor of s to be a factor of r, S ITSELF must be a factor of r.
It is not necessary that r and s have the same number of factors.
To illustrate:
If r=18 and s=6, every factor of s (1, 2, 3, 6) is also a factor of r (whose factors are 1, 2, 3, 6, 9, 18), but r has more factors than s.
Since s itself must be a factor of r, r/s must be an integer.
SUFFICIENT.

Statement 2: Every prime factor of s is also a prime factor of r.
Case 1: r=18 and s=6.
The prime factors of r=18 are 2 and 3.
The prime factors of s=6 are 2 and 3.
Thus, every prime factor of s is also a prime factor of r.
In this case, r/s = 18/6 = 3, which is an integer.

Case 2: r=6 and s=18.
The prime factors of r=6 are 2 and 3.
The prime factors of s=18 are 2 and 3.
Thus, every prime factor of s is also a prime factor of r.
In this case, r/s = 6/18 = 1/3, which is NOT an integer.
INSUFFICIENT.

The correct answer is A.
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by gander123 » Sun Jan 13, 2013 7:02 am
Thanks Mitch,

I got it ...

cheers,

Tobi
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by ceilidh.erickson » Tue Jan 15, 2013 10:01 am
Tobi, to elaborate further on your second question: yes, "every factor" does mean the same thing as "the same number/amount of factors." It's important to distinguish between the different ways that the GMAT asks about factors...

distinct factors / every factor: this is every number that divides evenly into another number, whether prime or not (but without counting any duplicates). For example, the distinct factors of 36 would be: 1, 2, 3, 4, 6, 9, 12, 18, 36

number of prime factors / prime factorization: You could think of this as "how many branches on the factor tree?", or is other words, when we express a number as a product of prime factors, how many of those prime factors are there? The prime factorization of 36 is 2^2 * 3^2, so there are 4 total primes factors: 2 * 2 * 3 * 3

distinct prime factors: Here, we just want to count the different prime factors, without counting any duplicates. For example, there are only 2 distinct prime factors of 36: 2 and 3. We don't want to count the duplicates.

So in statement (2) of this problem, we know that s and r share all of their distinct prime factors, but we don't know how many of each of those factors we have.
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by GMATGuruNY » Tue Jan 15, 2013 10:32 am
Just to clarify:

Every factor of s is also a factor of r does not mean that s and r have the same number of factors.
It means only that any positive integer that divides evenly into s must also divide evenly into r.
The result is that r must have AT LEAST AS MANY FACTORS as does s.
But it's possible for r to have many MORE FACTORS than does s.
For example, if r=60 and s=1, every factor of s is also a factor of r, but r=60 has many more factors than does s=1.
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by mcdesty » Sat Jul 19, 2014 9:15 pm
See Image below. For statement two, I could easily have found a Yes scenario.
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by GMATinsight » Sat Jul 19, 2014 9:36 pm
"If r and s are positive integers, is r/s an integer?

(1) Every factor of s is also a factor of r.

(2) Every prime factor of s is also a prime factor of r."
Question: Is r/s an Integer?

r/s will be an integer is s is a factor of r {both r and s are integers} therefore,

Question Rephrased : Is s a factor of r?

Statement 1) Every factor of s is also a factor of r
i.e. r is a multiple of s
i.e. s is a factor of r
SUFFICIENT

Statement 2) Every prime factor of s is also a prime factor of r
If s has power of prime factor higher than the power of same prime factor available in r then s will NOT be a factor of r (e.g. s= 2^3 and r = 2^2)
AND
If s has power of prime factors lower than the power of same prime factors available in r then s will be a factor of r (e.g. s= 2^2 and r = 2^3)
INSUFFICIENT

Answer: Option [spoiler]A[/spoiler]
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