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.
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Integers
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goalevan
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r, s > 0, integers
"Is r/s an integer?"
= "Is r a multiple of s?"
= "Is r divisible by s?"
= "Is s a factor of r?"
Statement 1) If every factor of s is a factor of r, then s is a factor of r.
For example, if s = (5)(6) = 30 and r = (30)(10)(5)(6) = 9000
r/s = (30)(10) = 300, which is an integer. Sufficient.
Statement 2) If every prime factor of s is a prime factor of r, then every factor of s is a factor of r, and s is a factor of r.
Every positive integer can be reduced to a factorization consisting only of prime numbers. From the above example we would have:
s = (2)(3)(5) = 30 and r = (2^3)(3^2)(5^3) = (2)(3)(5)(2^2)(3)(5^2)
s/r = (2^2)(3)(5^2) = 300, which is an integer. Sufficient.
D
"Is r/s an integer?"
= "Is r a multiple of s?"
= "Is r divisible by s?"
= "Is s a factor of r?"
Statement 1) If every factor of s is a factor of r, then s is a factor of r.
For example, if s = (5)(6) = 30 and r = (30)(10)(5)(6) = 9000
r/s = (30)(10) = 300, which is an integer. Sufficient.
Statement 2) If every prime factor of s is a prime factor of r, then every factor of s is a factor of r, and s is a factor of r.
Every positive integer can be reduced to a factorization consisting only of prime numbers. From the above example we would have:
s = (2)(3)(5) = 30 and r = (2^3)(3^2)(5^3) = (2)(3)(5)(2^2)(3)(5^2)
s/r = (2^2)(3)(5^2) = 300, which is an integer. Sufficient.
D
This is from a GMAT Prep practice test from the GMAT website, and this was the answer that I thought as well, however, the test said D was wrong and the correct answer was A......I just wanted to make sure that I wasn't crazy and that the test had an incorrect answer, thank you for your post.
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goalevan
- Master | Next Rank: 500 Posts
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For A to be correct, I would have thought it would be worded "every unique prime factor of s is also a unique prime factor of r" or "every different prime factor..."
But, I misinterpreted the statement, because the factors for a number, say 18, are all possible combinations of its prime factors, 2^1 * 3^2: 1, [2], [3], 6, 9, 18. The prime numbers only ever appear here once.
Thus:
Statement 2) If every prime factor of s is also a prime factor of r,
s could be 2^100 * 3^100 * 7^100
and r = 2 * 3 * 7
They would share the same prime factors:
r: 1, [2], [3], 6, [7], 14, 21, 42
s: 1, [2], [3], 4, 6, [7], 8, 9, 12, ...
r/s = 1/(2^99) * 1/(3^99) * 1/(7^99), definitely not an integer.
But, I misinterpreted the statement, because the factors for a number, say 18, are all possible combinations of its prime factors, 2^1 * 3^2: 1, [2], [3], 6, 9, 18. The prime numbers only ever appear here once.
Thus:
Statement 2) If every prime factor of s is also a prime factor of r,
s could be 2^100 * 3^100 * 7^100
and r = 2 * 3 * 7
They would share the same prime factors:
r: 1, [2], [3], 6, [7], 14, 21, 42
s: 1, [2], [3], 4, 6, [7], 8, 9, 12, ...
r/s = 1/(2^99) * 1/(3^99) * 1/(7^99), definitely not an integer.
Last edited by goalevan on Sun Jul 17, 2011 5:02 pm, edited 1 time in total.
