For a positive integer n, the function @n represents the product of al

Question

For a positive integer n, the function @n represents the product of all even numbers that lie between n and 2n, exclusive. For example, @6 = 8*10. If k is a positive integer and the greatest prime factor of @k is 17, which of the following cannot be the value of k?

I. 9
II. 18
III. 32

A. I only
B. II only
C. III only
D. I and III only
E. II and III only

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I. 9
II. 18
III. 32

A. I only
B. II only
C. III only
D. I and III only
E. II and III only

Are You Up For the Challenge: 700 Level Questions

ArunSharma12 · Answer

@n = product of even numbers between n and 2n.
@n must contain all even numbers.
for @k to have 17 as a prime factor, it must have 34 as one the numbers but should not include 38 included within  k .

K = 9: @k will not include 34. not possible
k = 18; @k range from 20 - 34. possible
k = 32; @k range from 34-62. not possible because it includes 38 which shifts greater prime factor to be more than 17 to 31   
ans: D

yashikaaggarwal · Answer

Since @n consist of the even number lies between n and 2n exclusive.
@k have 17 as the biggest prime number.
17 can't be the number itself because it's an odd number.
Therefore 34 is the number in the set.
@k = { .................34}
2K can be 36 or 38
Because even if 2k is 38 the largest no. will still be 17.
But 2k can't be 40. Because the greatest prime no. Will be 19 in that case.
2k = 36 or 38
K will be either 18 or 19
Only statement 2 is satisfying constraint
Therefore Answer should be B

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HoneyLemon · Answer

Ans is D .

@n = product of even numbers between n and 2n , exclusive n and 2n , and must contain all even numbers.

I. 9

-- @9 = 10 * 12 ......*16  not possible  ]

II. 18

@18  = 20 * 22 ......*34

III. 32
-- @32 = 34 * 12 ......*64   not possible  ]

Hence ans is D

firas92 · Answer

The question is which  cannot  be the value of k

From your solution k can be 18 but cannot be 9 or 32

So the answer is D

samarpan.g28 · Answer

I. If k=9 then the range is 9 to 18 -> 17 cannot be a factor of the product. Cancelled.
II. If k=18 then the range is 18 to 36 -> the maximum even integer in the range is 34=2*17. Keep this.
III. If k=32 then the range is 32 to 64 -> the maximum even integer in the range is 62=2*31. Cancelled.

I and III cannot be the value of k. Option (D) is correct.

wasario · Answer

My explanation with a tad bit more detail:

I) @9 = 10 * 12 * 14 * 16 (because 2n = 2 * 9 = 18)-> the prime factor of this product is not 17 at all, because none of the numbers here are divisible by 17. Since 9 cannot be k, this answer choice is correct
II) @18 = 20 * 22 * 24 * 26 * 28 * 30 * 32 * 34 (because 2n = 2 * 18 = 36) -> the biggest prime factor of this product actually IS 17, because 34 is divisible by 17, and there are no other prime products of a prime number and 2 in this sequence in which the prime number is bigger than 17. Since 18 is a valid value of k, this answer choice is incorrect.
III) @32 = 34 * 36 * 38 * 40 * ... * 62 (because 2n = 2 * 32 = 64) -> while 17 is a prime factor of this product due to the value of 34, it is NOT the biggest one. That would be 31, because of 62 (31 * 2) due to the same logic explained in II. Since 32 cannot be k, this answer choice is correct.

Choice I + III,  D  is the correct answer.

OJA · Answer

I think the key thing to remember is that the only even numbers needs to be included. Hence, we are looking for a set which includes 34 but not 38 because that would make the 19 to be the greatest prime number in the set.

For a positive integer n, the function @n represents the product of al

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For a positive integer n, the function @n represents the product of al

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