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How many different prime numbers are factors of the positive integer n?
1) Four different prime numbers are factors of 2n
2) Four different prime numbers are factors of n2
- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
How many different prime numbers are factors of the positive integer n...
1) Four different prime numbers are factors of 2n implies that one of the four prime numbers is 2 but 2 may be a factor of n, and if that is so then the total number of prime factors will not change.
If n = 2 * 3 * 5 * 7 (here n has 4 prime factors), then 2n = 2 * 2 * 3 * 5 * 7. Here 2n has 4 prime factors.
If n = 3 * 5 * 7 (here n has 3 prime factors), then 2n = 2 * 3 * 5 * 7. Here 2n has 4 prime factors.
From the above examples, it can be seen that number of different prime numbers may vary; NOT sufficient.
(2) Four different prime numbers are factors of n� implies four different prime numbers are factors of n as well; SUFFICIENT.
The correct answer is B.
Most Upvoted Answer
How many different prime numbers are factors of the positive integer n...
Answer:
Given: Positive integer n and its prime factors
To find: Number of different prime factors of n
Solution:
Statement 1: Four different prime numbers are factors of 2n
Let's consider the prime factorization of 2n: 2n = (2^k) * m, where m is an odd integer and k is a non-negative integer.
Since four different prime numbers are factors of 2n, we can say that at least two of these primes are factors of 2, i.e., 2 and another odd prime. Hence, k >= 1.
Now, let's consider the prime factorization of n: n = (2^j) * p, where p is an odd integer and j is a non-negative integer.
From statement 1, we know that the prime factors of 2n include at least two odd primes that are not factors of 2. These primes can only be factors of p, since p is the only odd integer that appears in the prime factorization of n.
Therefore, the number of different prime factors of n is at least 2.
However, statement 1 does not give us any information about the number of prime factors of n beyond that it is at least 2. Therefore, statement 1 alone is not sufficient to answer the question.
Statement 2: Four different prime numbers are factors of n
Let's consider the prime factorization of n: n = (2^j) * p, where p is an odd integer and j is a non-negative integer.
From statement 2, we know that n has four different prime factors. Since 2 is the only even prime, we can say that p has at least three different prime factors.
Therefore, the number of different prime factors of n is at least 3.
However, statement 2 does not give us any information about the specific primes that are factors of n. Therefore, statement 2 alone is not sufficient to answer the question.
Combining statements 1 and 2:
From statement 1, we know that at least two of the odd primes that are factors of 2n are also factors of n. From statement 2, we know that n has at least three different prime factors.
Therefore, the number of different prime factors of n is at least 2 (from statement 1) and at least 3 (from statement 2), which means that the number of different prime factors of n is exactly 3.
Therefore, statement 1 and statement 2 together are sufficient to answer the question.
Answer: A
Given: Positive integer n and its prime factors
To find: Number of different prime factors of n
Solution:
Statement 1: Four different prime numbers are factors of 2n
Let's consider the prime factorization of 2n: 2n = (2^k) * m, where m is an odd integer and k is a non-negative integer.
Since four different prime numbers are factors of 2n, we can say that at least two of these primes are factors of 2, i.e., 2 and another odd prime. Hence, k >= 1.
Now, let's consider the prime factorization of n: n = (2^j) * p, where p is an odd integer and j is a non-negative integer.
From statement 1, we know that the prime factors of 2n include at least two odd primes that are not factors of 2. These primes can only be factors of p, since p is the only odd integer that appears in the prime factorization of n.
Therefore, the number of different prime factors of n is at least 2.
However, statement 1 does not give us any information about the number of prime factors of n beyond that it is at least 2. Therefore, statement 1 alone is not sufficient to answer the question.
Statement 2: Four different prime numbers are factors of n
Let's consider the prime factorization of n: n = (2^j) * p, where p is an odd integer and j is a non-negative integer.
From statement 2, we know that n has four different prime factors. Since 2 is the only even prime, we can say that p has at least three different prime factors.
Therefore, the number of different prime factors of n is at least 3.
However, statement 2 does not give us any information about the specific primes that are factors of n. Therefore, statement 2 alone is not sufficient to answer the question.
Combining statements 1 and 2:
From statement 1, we know that at least two of the odd primes that are factors of 2n are also factors of n. From statement 2, we know that n has at least three different prime factors.
Therefore, the number of different prime factors of n is at least 2 (from statement 1) and at least 3 (from statement 2), which means that the number of different prime factors of n is exactly 3.
Therefore, statement 1 and statement 2 together are sufficient to answer the question.
Answer: A
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How many different prime numbers are factors of the positive integer n?1) Four different prime numbers are factors of 2n2) Four different prime numbers are factors of n2a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'A'. Can you explain this answer?
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How many different prime numbers are factors of the positive integer n?1) Four different prime numbers are factors of 2n2) Four different prime numbers are factors of n2a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'A'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about How many different prime numbers are factors of the positive integer n?1) Four different prime numbers are factors of 2n2) Four different prime numbers are factors of n2a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many different prime numbers are factors of the positive integer n?1) Four different prime numbers are factors of 2n2) Four different prime numbers are factors of n2a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'A'. Can you explain this answer?.
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