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What is the number of positive integers that divide k but do not divide k, where k is a positive integer?
(1) k2 has a total of 13 factors
(2) √k has a total of 4 factors
(1) k2 has a total of 13 factors
(2) √k has a total of 4 factors
- a)Statement (1) ALONE is sufficient, but statement (2) alone is
not sufficient to answer the question asked. - b)Statement (2) ALONE is sufficient, but statement (1) alone is
not sufficient to answer the question asked. - c)BOTH statements (1) and (2) TOGETHER are sufficient to
answer the question asked, but NEITHER statement ALONE
is sufficient to answer the question asked. - d)EACH statement ALONE is sufficient to answer the question
asked. - e)Statements (1) and (2) TOGETHER are NOT sufficient to
answer the question asked, and additional data specific to the
problem are needed.
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
What is the number of positive integers that divide k but do not divid...
Step 1 & 2: Understand Question and Draw Inference
To Find: number of positive integers that divide k2 but do not divide k
- Number of integers that divide k2 = factors of k2
- Now, all the integers that are a factor of k must be the factors of k2 (As k is a factor of k2 )
- So, the factors of k2 that do not divide k = Factors(k2 ) - Factors(k)
- Factors (k)= (a+1)(b+1)….
- Factors (k2 = (2a +1) *(2b +1)….
- So, we need to find the value of: (2a +1) *(2b +1)…. - (a+1)(b+1)….
Step 3 : Analyze Statement 1 independent
1. k2 has a total of 13 factors
- Number of Factors (k2 = (2a+1)(2b+1)… = 13.
- As 13 is a prime number, there is only one way to express 13 as a product of two or more positive integers, i.e. 13 * 1
- So, (2a+1)*(2b+1)… = 13* 1
- So, 2a+1 = 13 and 2b + 1= 1
- a = 6 and b = 0
- Thus k = P16 and k2 = P112
- Number of Factors (k) = 7 and Factors (k2) = 13
Hence, there are a total of 6 positive integers that divide k but do not divide k. Sufficient to answer
Step 4 : Analyze Statement 2 independent
1. √k has a total of 4 factors
- Now 4 can be expressed as a product of two or more positive integers in the following possible ways:
- Thus, there will be 25 – 9 = 16 factors that will divide k2 but will not divide k
As we do not have a unique answer, this statement is insufficient to answer.
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step-3, this step is not required
Answer A
Most Upvoted Answer
What is the number of positive integers that divide k but do not divid...
Statement (1): k^2 has a total of 13 factors
To determine the number of positive integers that divide k but do not divide k, we need to find the prime factorization of k. Let's assume the prime factorization of k is given by:
k = p1^a1 * p2^a2 * p3^a3 * ... * pn^an
Where p1, p2, p3, ..., pn are distinct prime numbers and a1, a2, a3, ..., an are positive integers.
From statement (1), we know that k^2 has a total of 13 factors. The total number of factors of k^2 can be determined using the formula:
Total number of factors = (a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1)
Since k^2 has 13 factors, we can write the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 13
We need to find the possible values of a1, a2, a3, ..., an that satisfy this equation.
Statement (2): k has a total of 4 factors
From statement (2), we know that k has a total of 4 factors. The total number of factors of k can be determined using the formula:
Total number of factors = (a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1)
Since k has 4 factors, we can write the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 4
We need to find the possible values of a1, a2, a3, ..., an that satisfy this equation.
Combining both statements:
From statement (1), we have the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 13
From statement (2), we have the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 4
Since both statements provide equations that determine the possible values of a1, a2, a3, ..., an, we can combine the equations and solve for the values.
If we solve the combined equations, we will find that the only possible values of a1, a2, a3, ..., an that satisfy both equations are 2, 2, 2, 2. This means that k = p1^2 * p2^2 * p3^2 * p4^2, where p1, p2, p3, and p4 are distinct prime numbers.
Therefore, combining both statements is sufficient to determine the prime factorization of k, and subsequently, the number of positive integers that divide k but do not divide k.
Hence, the correct answer is option A.
To determine the number of positive integers that divide k but do not divide k, we need to find the prime factorization of k. Let's assume the prime factorization of k is given by:
k = p1^a1 * p2^a2 * p3^a3 * ... * pn^an
Where p1, p2, p3, ..., pn are distinct prime numbers and a1, a2, a3, ..., an are positive integers.
From statement (1), we know that k^2 has a total of 13 factors. The total number of factors of k^2 can be determined using the formula:
Total number of factors = (a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1)
Since k^2 has 13 factors, we can write the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 13
We need to find the possible values of a1, a2, a3, ..., an that satisfy this equation.
Statement (2): k has a total of 4 factors
From statement (2), we know that k has a total of 4 factors. The total number of factors of k can be determined using the formula:
Total number of factors = (a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1)
Since k has 4 factors, we can write the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 4
We need to find the possible values of a1, a2, a3, ..., an that satisfy this equation.
Combining both statements:
From statement (1), we have the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 13
From statement (2), we have the equation:
(a1 + 1)*(a2 + 1)*(a3 + 1)*...*(an + 1) = 4
Since both statements provide equations that determine the possible values of a1, a2, a3, ..., an, we can combine the equations and solve for the values.
If we solve the combined equations, we will find that the only possible values of a1, a2, a3, ..., an that satisfy both equations are 2, 2, 2, 2. This means that k = p1^2 * p2^2 * p3^2 * p4^2, where p1, p2, p3, and p4 are distinct prime numbers.
Therefore, combining both statements is sufficient to determine the prime factorization of k, and subsequently, the number of positive integers that divide k but do not divide k.
Hence, the correct answer is option A.
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What is the number of positive integers that divide k but do not divide k, where k is a positive integer?(1) k2 has a total of 13 factors(2) √k has a total of 4 factorsa)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?
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What is the number of positive integers that divide k but do not divide k, where k is a positive integer?(1) k2 has a total of 13 factors(2) √k has a total of 4 factorsa)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct 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 What is the number of positive integers that divide k but do not divide k, where k is a positive integer?(1) k2 has a total of 13 factors(2) √k has a total of 4 factorsa)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct 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 What is the number of positive integers that divide k but do not divide k, where k is a positive integer?(1) k2 has a total of 13 factors(2) √k has a total of 4 factorsa)Statement (1) ALONE is sufficient, but statement (2) alone isnot sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone isnot sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient toanswer the question asked, but NEITHER statement ALONEis sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the questionasked.e)Statements (1) and (2) TOGETHER are NOT sufficient toanswer the question asked, and additional data specific to theproblem are needed.Correct answer is option 'A'. Can you explain this answer?.
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