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If x is a positive integer, is x5 + 1 an odd number?
(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
(2) 5x is an odd number
(2) 5x is an odd number
- 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
- 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 are needed
Correct answer is option 'A'. Can you explain this answer?
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If x is a positive integer, is x5 + 1 an odd number?(1) x is the small...
Statement (1): x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
To determine if x^5 - 1 is odd, we need to know the value of x.
If x is an even number, then x^5 will also be even, and subtracting 1 will result in an odd number. However, if x is an odd number, then x^5 will also be odd, and subtracting 1 will result in an even number.
From statement (1), we know that x is divisible by all integers from 21 to 24. Let's examine these numbers:
21 = 3 * 7
22 = 2 * 11
23 = prime number
24 = 2^3 * 3
The smallest integer that is divisible by all of these numbers is the least common multiple (LCM) of these numbers, which is 2^3 * 3 * 7 * 11 = 2^3 * 3 * 7 * 11.
Since x is the smallest integer divisible by these numbers, we can conclude that x must be equal to 2^3 * 3 * 7 * 11.
Statement (2): 5x is an odd number
If 5x is an odd number, then x must be an odd number.
From statement (1), we found that x is equal to 2^3 * 3 * 7 * 11, which is an even number. Therefore, statement (2) is not consistent with statement (1).
Combining Statements (1) and (2):
Since statement (2) contradicts statement (1), the statements cannot be combined to provide a consistent answer. Therefore, the correct answer is option A - Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.
To determine if x^5 - 1 is odd, we need to know the value of x.
If x is an even number, then x^5 will also be even, and subtracting 1 will result in an odd number. However, if x is an odd number, then x^5 will also be odd, and subtracting 1 will result in an even number.
From statement (1), we know that x is divisible by all integers from 21 to 24. Let's examine these numbers:
21 = 3 * 7
22 = 2 * 11
23 = prime number
24 = 2^3 * 3
The smallest integer that is divisible by all of these numbers is the least common multiple (LCM) of these numbers, which is 2^3 * 3 * 7 * 11 = 2^3 * 3 * 7 * 11.
Since x is the smallest integer divisible by these numbers, we can conclude that x must be equal to 2^3 * 3 * 7 * 11.
Statement (2): 5x is an odd number
If 5x is an odd number, then x must be an odd number.
From statement (1), we found that x is equal to 2^3 * 3 * 7 * 11, which is an even number. Therefore, statement (2) is not consistent with statement (1).
Combining Statements (1) and (2):
Since statement (2) contradicts statement (1), the statements cannot be combined to provide a consistent answer. Therefore, the correct answer is option A - Statement (1) alone is sufficient, but statement (2) alone is not sufficient to answer the question asked.
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If x is a positive integer, is x5 + 1 an odd number?(1) x is the small...
Statement (1): x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.
If x is the smallest integer divisible by all integers from 21 to 24, it means that x is a multiple of their least common multiple (LCM). The LCM of 21, 22, 23, and 24 is 24. Therefore, x must be a multiple of 24. Let's consider a few values of x:
If x is the smallest integer divisible by all integers from 21 to 24, it means that x is a multiple of their least common multiple (LCM). The LCM of 21, 22, 23, and 24 is 24. Therefore, x must be a multiple of 24. Let's consider a few values of x:
If x = 24, then x5 + 1 = 245 + 1 = 796,594,177, which is an odd number.
If x = 48, then x5 + 1 = 485 + 1 = 65,611,007,681, which is an odd number.
Based on the above examples, it seems that x5 + 1 is always an odd number when x is a multiple of 24. However, we cannot be certain that it holds true for all positive integers. Therefore, statement (1) alone is not sufficient to answer the question.
If x = 48, then x5 + 1 = 485 + 1 = 65,611,007,681, which is an odd number.
Based on the above examples, it seems that x5 + 1 is always an odd number when x is a multiple of 24. However, we cannot be certain that it holds true for all positive integers. Therefore, statement (1) alone is not sufficient to answer the question.
Statement (2): 5x is an odd number.
If 5x is an odd number, it means that x must be an odd number as well since an even number multiplied by an odd number is always even. However, knowing that x is odd does not provide enough information to determine if x5 + 1 is an odd number. For example, if x = 3, then x5 + 1 = 35 + 1 = 244, which is an even number. Therefore, statement (2) alone is not sufficient to answer the question.
If 5x is an odd number, it means that x must be an odd number as well since an even number multiplied by an odd number is always even. However, knowing that x is odd does not provide enough information to determine if x5 + 1 is an odd number. For example, if x = 3, then x5 + 1 = 35 + 1 = 244, which is an even number. Therefore, statement (2) alone is not sufficient to answer the question.
Combining both statements, we know that x is a positive integer that is divisible by all integers from 21 to 24, and 5x is an odd number. Since x must be a multiple of 24, we can substitute x = 24k, where k is a positive integer, into the expression x5 + 1:
x5 + 1 = (24k)5 + 1 = 24,000,000k5 + 1 = 2(12,000,000k5) + 1.
The expression 2(12,000,000k5) is always even, and adding 1 to an even number results in an odd number. Therefore, x5 + 1 is always an odd number when x is a multiple of 24.
Since statement (1) alone is sufficient to determine that x is a multiple of 24, and we have concluded that x5 + 1 is always an odd number when x is a multiple of 24, the answer is option A: Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
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If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'A'. Can you explain this answer?
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If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect 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 If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect 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 If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'A'. Can you explain this answer?.
If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect 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 If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect 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 If x is a positive integer, is x5 + 1 an odd number?(1) x is the smallest integer that is divisible by all integers from 21 to 24, inclusive.(2) 5x is an odd numbera)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question askedb)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question askedc)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficientd)EACH statement ALONE is sufficient to answer the question askede)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are neededCorrect answer is option 'A'. Can you explain this answer?.
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