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For an integer n greater than 1, n! denotes the product of all integers from 1 to n, inclusive. If x and y are two distinct positive integers such that y > x, what are the values of x and y?
(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.
(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectively
  • 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 'C'. Can you explain this answer?

Answers

Step 1 & 2: Understand Question and Draw Inference
  • n! = 1*2*3……*n
  • x, y are distinct integers > o
  • y > x
To Find: Values of x and y
Step 3 : Analyze Statement 1 independent
  1. The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.
  • x! = 1*2*3…..*x
  • y! = 1*2*3….x*(x+1)…….* y ( as y > x)
    • So, we can write: y! = x! * a, where a is a positive integer
  • As y! is a multiple of x!, LCM( x!, y!) = y!
  • Similarly, as x! is a factor of y!, GCD(x!, y!) = x!
So,
Hence, 20 needs to be expressed as a product of 1 or more consecutive positive
integers. Following cases are possible:
  • As 20 = 22 * 5, following cases are possible:
  • 20 = 4 * 5
    • So, (x+1) * y = 20. Therefore, x + 1 = 4, i.e. x = 3 and y = 5
  • The other possible case is when (x+1) = y = 20, i.e. y = 20 and x = 19. So,
As we do not have unique values of x and y, the statement is insufficient to answer
Step 4 : Analyze Statement 2 independent
2. The lowest common multiple and the highest common factor of y! and x!
have 3 and 2 prime factors respectively.
  • we know that LCM(x!, y!) = y! = 1*2*3*4*5…y (we deduced this in the analysis of statement-1)
    • As y! has 3 prime factors only, they have to be the smallest 3 prime factors, i.e. {2, 3 , 5}. So, 5! ≤ y! < 7! as 7! will have 4 prime factors {2, 3, 5, 7}
    • 5 ≤ y < 7. So, y = 5 or 6
  • We also know that GCD(x!, y!) = x! = 1*2*3… x (we deduced this in the analysis of statement-1)
    • As x! has 2 prime factors only, they have to be the smallest 2 prime factors , i.e. {2, 3}. So, 3!≤ x! < 5!, as 5! will have 3 prime factors {2, 3, 5}
    • 3≤ x < 5 So, x = {3, 4}
As we do not have unique values of x and y, insufficient to answer
Step 5: Analyze Both Statements Together (if needed)
1. From statement-1, we inferred that y = {20, 5} and x = {19, 3}
2. From statement-2, we inferred that y = { 5, 6} and x = { 3,4 }
Combining both the statements, we have y = 5 and x = 3.
Sufficient to answer.
Answer: C

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Question Description
For an integer n greater than 1, n! denotes the product of all integers from 1 to n, inclusive. If x and y are two distinct positive integers such that y > x, what are the values of x and y?(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectivelya)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 'C'. Can you explain this answer? for GMAT 2023 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about For an integer n greater than 1, n! denotes the product of all integers from 1 to n, inclusive. If x and y are two distinct positive integers such that y > x, what are the values of x and y?(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectivelya)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 'C'. Can you explain this answer? covers all topics & solutions for GMAT 2023 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For an integer n greater than 1, n! denotes the product of all integers from 1 to n, inclusive. If x and y are two distinct positive integers such that y > x, what are the values of x and y?(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectivelya)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 'C'. Can you explain this answer?.
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If x and y are two distinct positive integers such that y > x, what are the values of x and y?(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectivelya)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 'C'. 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Can you explain this answer? theory, EduRev gives you an ample number of questions to practice For an integer n greater than 1, n! denotes the product of all integers from 1 to n, inclusive. If x and y are two distinct positive integers such that y > x, what are the values of x and y?(1) The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1.(2) The lowest common multiple and the highest common factor of y! and x! have 3 and 2 prime factors respectivelya)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 'C'. Can you explain this answer? tests, examples and also practice GMAT tests.
Step 1 2: Understand Question and Draw Inference n! = 1*2*3*n x, y are distinct integers o y xTo Find: Values of x and yStep 3 : Analyze Statement 1 independent The ratio of the lowest common multiple and the highest common factor of y! and x! is 20:1. x! = 1*2*3..*x y! = 1*2*3.x*(x+1).* y ( as y x) So, we can write: y! = x! * a, where a is a positive integer As y! is a multiple of x!, LCM( x!, y!) = y! Similarly, as x! is a factor of y!, GCD(x!, y!) = x!So,Hence, 20 needs to be expressed as a product of 1 or more consecutive positiveintegers. Following cases are possible: As 20 = 22 * 5, following cases are possible: 20 = 4 * 5 So, (x+1) * y = 20. Therefore, x + 1 = 4, i.e. x = 3 and y = 5 The other possible case is when (x+1) = y = 20, i.e. y = 20 and x = 19. So,As we do not have unique values of x and y, the statement is insufficient to answerStep 4 : Analyze Statement 2 independent2. The lowest common multiple and the highest common factor of y! and x!have 3 and 2 prime factors respectively. we know that LCM(x!, y!) = y! = 1*2*3*4*5y (we deduced this in the analysis of statement-1) As y! has 3 prime factors only, they have to be the smallest 3 prime factors, i.e. {2, 3 , 5}. So, 5! y! 7! as 7! will have 4 prime factors {2, 3, 5, 7} 5 y 7. So, y = 5 or 6 We also know that GCD(x!, y!) = x! = 1*2*3 x (we deduced this in the analysis of statement-1) As x! has 2 prime factors only, they have to be the smallest 2 prime factors , i.e. {2, 3}. So, 3! x! 5!, as 5! will have 3 prime factors {2, 3, 5} 3 x 5 So, x = {3, 4} As we do not have unique values of x and y, insufficient to answerStep 5: Analyze Both Statements Together (if needed)1. From statement-1, we inferred that y = {20, 5} and x = {19, 3}2. From statement-2, we inferred that y = { 5, 6} and x = { 3,4 }Combining both the statements, we have y = 5 and x = 3.Sufficient to answer.Answer: C