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21 Mar 2019, 21:47
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C
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38% (02:04) correct
62%
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wrong
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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.
(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.
21 Mar 2019, 22:43
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This is a very good question for understanding the concepts of factorials, LCM and HCF.
WhatsApp Image 2019-03-22 at 11.12.13.jpeg [ 72.36 KiB | Viewed 4269 times ]
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WhatsApp Image 2019-03-22 at 11.12.13.jpeg [ 72.36 KiB | Viewed 4269 times ]
06 Jun 2019, 10:30
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Any larger factorial is divisible by every smaller factorial, so if y > x, then y! is divisible by x!. So the LCM of y! and x! will just be y!, and the GCD of y! and x! will just be x!. So Statement 1 tells us y!/x! = 20/1, or y! = 20x!. That can be true in two situations: since 5! = (5)(4)(3!) = 20(3!), we might have y=5 and x=3. And since 20! = (20)(19!), we might have y=20 and x=19. So Statement 1 is not sufficient.
Statement 2 tells us that y! has three prime factors, and x! has two prime factors. So y! must be divisible by 2, 3 and 5, and no other primes (since y! is always divisible by every prime up to y), and y can be 5 or 6, and x! must be divisible by 2 and 3, and x can be 3 or 4. So Statement 2 is not sufficient.
Combined, the only possibility is that y = 5 and x = 3, so the answer is C.
Statement 2 tells us that y! has three prime factors, and x! has two prime factors. So y! must be divisible by 2, 3 and 5, and no other primes (since y! is always divisible by every prime up to y), and y can be 5 or 6, and x! must be divisible by 2 and 3, and x can be 3 or 4. So Statement 2 is not sufficient.
Combined, the only possibility is that y = 5 and x = 3, so the answer is C.
