Quotient of positive integers r and s

Question

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

GMATGuruNY · Answer

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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gander123 · Answer

Thanks Mitch, I got it ... cheers, Tobi

ceilidh.erickson · Answer

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.

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

GMATGuruNY · Answer

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.

mcdesty · Answer

See Image below. For statement two, I could easily have found a Yes scenario.

GMATinsight · Answer

"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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