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17 May 2019, 06:34
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A
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Difficulty:
45%
(medium)
Question Stats:
57% (01:06) correct
43%
(01:15)
wrong
based on 180
sessions
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If p and q are positive integers, is p/q also an integer?
I. Every factor of q is also a factor of p.
II. Every prime factor of q is also a prime factor of p.
I. Every factor of q is also a factor of p.
II. Every prime factor of q is also a prime factor of p.
21 May 2019, 06:47
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This is a question that tests you on your knowledge of a finer aspect of positive integers. It’s also a question where your knowledge of prime factorization can help you interpret the second statement better, without falling for the trap answer.
Any positive integer, except 1, will always have 2 factors – 1 and the number itself. If it has only these two numbers as factors, it is termed a prime number; if it has some more numbers as factors, apart from these two, it is called a composite number.
You are already aware of the above concepts, from our posts on Prime numbers.
From statement I alone, we know that every factor of q is also a factor of p. From the question data, we know that both p and q are positive integers.
So, one of the factors of q, will be q itself. Therefore, if we say ‘every factor of q is also a factor of p’, we are essentially saying that q is also a factor of p.
Since q is a factor of p, p/q HAS to be an integer. So, statement I alone is sufficient. Hence, the possible answer options are A or D and answer options B, C and E can be ruled out.
From statement II alone, we know that q and p have the same prime factors. But, we do now know anything about the powers to which these prime factors are raised, in p and q respectively. For example,
If we assume that p and q both have 2 and 3 as their prime factors,
Let p = \(2 ^2\) * \(3^2\) and q = 2 * 3. Here, p = 36 and q = 6. p/q is definitely an integer.
When p = 2 * 3 and q = \(2^2\) * \(3^2\), p/q will not be an integer.
Therefore, statement II alone is insufficient. Answer option D gets eliminated and the correct answer option is A.
While interpreting statements like statement II, it is important to know how to distinguish between the factor of a number and the prime factor of a number. Not to forget that prime factors of a number can be raised to different powers, which have to be factored in while analyzing the data.
Hope this helps!
Any positive integer, except 1, will always have 2 factors – 1 and the number itself. If it has only these two numbers as factors, it is termed a prime number; if it has some more numbers as factors, apart from these two, it is called a composite number.
You are already aware of the above concepts, from our posts on Prime numbers.
From statement I alone, we know that every factor of q is also a factor of p. From the question data, we know that both p and q are positive integers.
So, one of the factors of q, will be q itself. Therefore, if we say ‘every factor of q is also a factor of p’, we are essentially saying that q is also a factor of p.
Since q is a factor of p, p/q HAS to be an integer. So, statement I alone is sufficient. Hence, the possible answer options are A or D and answer options B, C and E can be ruled out.
From statement II alone, we know that q and p have the same prime factors. But, we do now know anything about the powers to which these prime factors are raised, in p and q respectively. For example,
If we assume that p and q both have 2 and 3 as their prime factors,
Let p = \(2 ^2\) * \(3^2\) and q = 2 * 3. Here, p = 36 and q = 6. p/q is definitely an integer.
When p = 2 * 3 and q = \(2^2\) * \(3^2\), p/q will not be an integer.
Therefore, statement II alone is insufficient. Answer option D gets eliminated and the correct answer option is A.
While interpreting statements like statement II, it is important to know how to distinguish between the factor of a number and the prime factor of a number. Not to forget that prime factors of a number can be raised to different powers, which have to be factored in while analyzing the data.
Hope this helps!
General Discussion
17 May 2019, 08:53
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The post above doesn't cite the source of the problem, but this is an old official question, one that has appeared in GMATPrep tests (with the letters changed from 'r' and 's' to 'p' and 'q') -
https://gmatclub.com/forum/if-r-and-s-a ... 28005.html
https://gmatclub.com/forum/if-r-and-s-a ... 28005.html
