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If x is an integer, does x have a factor n such that 1 <
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08 Sep 2010, 11:52
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If x is an integer, does x have a factor n such that 1 < n < x? (1) x > 3! (2) 15! + 2 ≤ x ≤ 15! + 15 I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............
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If x is an integer, does x have a factor n such that 1 <
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08 Sep 2010, 12:01
zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) cannot be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B.
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Re: factor factorials
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08 Sep 2010, 17:54
I love this question, so I'll chime in even if only to bump it to the top so more people see it! Hopefully most can see pretty quickly that x > 3! just means x > 6, and that isn't nearly enough to tell us whether it is prime. Statement 2 is pretty neat, though: 15! + 2 ≤ x ≤ 15! + 15 Think about 15!. 15! is 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1, which means that: 15! is a multiple of 2 15! is a multiple of 3 15! is a multiple of 4 etc.... 15! is a multiple of 15 Now think about multiples of 2. Every SECOND number is a multiple of 2: 2, 4, 6, 8, 10. You create multiples of 2 by adding 2 to a previous multiple of 2. If a number is even, adding 2 "keeps it" even. The same holds for 3. Every THIRD number (3, 6, 9, 12, 15, 18) is a multiple of 3. If you have a multiple of 3 and add 3 to it, it's still a multiple of 3. This will hold for all of these numbers  because 15! is a multiple of every number between 2 and 15, then adding any number in that range of 215 will ensure that the new number remains a multiple of that number, and the new number will not be prime. Therefore, statement 2 is sufficient, and the correct answer is B. Now...consider the number 15! + 1. It's too big a number to know offhand whether it's prime, but we do know that it is NOT a multiple of any numbers 215. In order to be a multiple of 2, we'd have to add a multiple of 2 to 15!, and 1 breaks us off of that everysecondnumber cycle. Same for 3  we'd need to add a multiple of 3 in order to keep 15! on that everythird cycle, so 15! is not a multiple of 3. We can prove that 15! + 1 is not divisible by any numbers between 2 and 15. It's largest prime factor must then be 17 or greater. Understanding that ideology and being able to determine divisibility of large numbers can be quite helpful on questions that might otherwise seem impossible. Thanks for posting this question!
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Re: factor factorials
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08 Sep 2010, 13:17
zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ Rephrase: Is there a factor of x that is greater than 1? So x should not be prime 1. if x> 3! then x>6 but 7 is a prime and greater than 6 and 9 is not prine but greater than 6. No solid answer 2. If x lies between 15!+2 and 15!+ 15 then there are no primes here.. B



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Re: factor factorials
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09 Sep 2010, 06:26
Nice rephrase. Somehow in option 1 I took my values as 3!, 4!, 5! and thought they all will have a factor as needed.
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Re: factor factorials
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09 Sep 2010, 14:28
Bunuel wrote: zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) can not be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B. I am able to follow until the point that I have highlighted red....... tried to work your point my self so decided to find if x for 5!+3<x<5!+8 works with your statement above.... so  \(5! = 120\) \(5!+3 / 3 = 123 / 3 =41\)correct \(5!+4 / 4 = 124 / 4 =31\) correct \(5!+5 / 5 = 125 / 5 =25\) correct \(5!+6 / 6 = 126 / 6 =21\)correct \(5!+7 / 7 = 127 / 7 =18.14\)INCORRECT !!!!so...? what am i doing wrong???



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If x is an integer, does x have a factor n such that 1 <
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09 Sep 2010, 14:37
zisis wrote: I am able to follow until the point that I have highlighted red....... tried to work your point my self so decided to find if x for 5!+3<x<5!+8 works with your statement above....
so  \(5! = 120\) \(5!+3 / 3 = 123 / 3 =41\)correct \(5!+4 / 4 = 124 / 4 =31\) correct \(5!+5 / 5 = 125 / 5 =25\) correct \(5!+6 / 6 = 126 / 6 =21\)correct \(5!+7 / 7 = 127 / 7 =18.14\)INCORRECT !!!!
so...? what am i doing wrong??? You replaced 15! by 5!. Thus you cannot factor out 7 out of 5!+8 and this is the whole point here. If you want to check with smaller numbers try \(5!+2\leq{x}\leq{5!+5}\)
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Re: factor factorials
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09 Sep 2010, 15:49
Bunuel wrote: zisis wrote: I am able to follow until the point that I have highlighted red....... tried to work your point my self so decided to find if x for 5!+3<x<5!+8 works with your statement above....
so  \(5! = 120\) \(5!+3 / 3 = 123 / 3 =41\)correct \(5!+4 / 4 = 124 / 4 =31\) correct \(5!+5 / 5 = 125 / 5 =25\) correct \(5!+6 / 6 = 126 / 6 =21\)correct \(5!+7 / 7 = 127 / 7 =18.14\)INCORRECT !!!!
so...? what am i doing wrong??? You replaced 15! by 5!. Thus you can not factor out 7 out of 5!+8 and this is the whole point here. did it on excel and seems like you are right..... 15! 15!+x (15!+x)/x 1,307,674,368,000.00 1,307,674,368,003.00 435,891,456,001.00 1,307,674,368,000.00 1,307,674,368,004.00 326,918,592,001.00 1,307,674,368,000.00 1,307,674,368,005.00 261,534,873,601.00 1,307,674,368,000.00 1,307,674,368,006.00 217,945,728,001.00 1,307,674,368,000.00 1,307,674,368,007.00 186,810,624,001.00 1,307,674,368,000.00 1,307,674,368,008.00 163,459,296,001.00 now have to go back and understand how it works.....using the example taht I used (which you mentioned I should use smaller numbers), how do you know when x is too big for a and b, when k!+a<x<k!+b ?



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Re: factor factorials
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09 Sep 2010, 16:19
zisis wrote: Bunuel wrote: zisis wrote: I am able to follow until the point that I have highlighted red....... tried to work your point my self so decided to find if x for 5!+3<x<5!+8 works with your statement above....
so  \(5! = 120\) \(5!+3 / 3 = 123 / 3 =41\)correct \(5!+4 / 4 = 124 / 4 =31\) correct \(5!+5 / 5 = 125 / 5 =25\) correct \(5!+6 / 6 = 126 / 6 =21\)correct \(5!+7 / 7 = 127 / 7 =18.14\)INCORRECT !!!!
so...? what am i doing wrong??? You replaced 15! by 5!. Thus you can not factor out 7 out of 5!+8 and this is the whole point here. did it on excel and seems like you are right..... 15! 15!+x (15!+x)/x 1,307,674,368,000.00 1,307,674,368,003.00 435,891,456,001.00 1,307,674,368,000.00 1,307,674,368,004.00 326,918,592,001.00 1,307,674,368,000.00 1,307,674,368,005.00 261,534,873,601.00 1,307,674,368,000.00 1,307,674,368,006.00 217,945,728,001.00 1,307,674,368,000.00 1,307,674,368,007.00 186,810,624,001.00 1,307,674,368,000.00 1,307,674,368,008.00 163,459,296,001.00 now have to go back and understand how it works.....using the example taht I used (which you mentioned I should use smaller numbers), how do you know when x is too big for a and b, when k!+a<x<k!+b ? It seems that you don't understand the explanation. No need to check in excel: no integer \(x\) satisfying \(15!+2\leq{x}\leq{15!+15}\) will be a prime number. There are 14 numbers satisfying it: If \(x=15!+2\) then we can factor out 2, so \(x\) would be multiple of 2, thus not a prime; If \(x=15!+3\) then we can factor out 3, so \(x\) would be multiple of 3, thus not a prime; ... If \(x=15!+15\) then we can factor out 15, so \(x\) would be multiple of 15, thus not a prime. Also 15!+{any multiple of prime less than or equal to 13} also won't be a prime number as we can factor out this prime (15! has all primes less than or equal to 13). Now, for \(x=15!+1\) or \(x=15!+17\) or \(x=15!+19\) we can not say for sure whether they are primes or not. In fact they are such a huge numbers that without a computer it's very hard and time consuming to varify their primality. Hope it's clear.
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Re: If x is an integer, does x have a factor n such that 1 <
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04 May 2014, 05:45
good qs!! and explainaton bunnel i am stalking you on GMATclub.com
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Re: If x is an integer, does x have a factor n such that 1 <
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21 Feb 2017, 06:33
Bunuel wrote: zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) can not be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B. Instead of multiplying all the numbers from 1 to 15, why not just take prime numbers between 1 and 15. That should still work, right?



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Re: If x is an integer, does x have a factor n such that 1 <
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21 Feb 2017, 10:55
SVP482 wrote: Bunuel wrote: zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) can not be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B. Instead of multiplying all the numbers from 1 to 15, why not just take prime numbers between 1 and 15. That should still work, right? No, it won't. Why should it? There are primes as well as nonprimes from 1 to 15.
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If x is an integer, does x have a factor n such that 1 <
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29 Jun 2017, 15:27
Given: x is an integer Question: Does x has a factor n such that 1<n<x? > This a 'yes' or 'no' question
If x is not a prime number, then n (factor of x) should always be less equal than x. Example: x = 8, factors of x i.e. n can be 1,2,4,8.
If x is a prime number, then n (factor of x) should always be 1, x Example: x = 3, factors of x i.e n can be 1, 3
Statement 1: x > 4! => 4*3*2*1 = 24 now factors of x can be 1, 2, 3, 4, 6, 8, 12, 24. For the values of 1,2,3,4,6,12 it satisfies the condition we get 'yes' for the question but for 24 we get 'no'. Therefore, it is (not sufficient).
Statement 2: 15! + 2 ≤ x ≤ 15! + 15 > range of value for x
15! = 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 15! + 2 = 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1 + 2 = 2(15*14*13*12*11*10*9*8*7*6*5*4*3*1 + 1) > is completely divisible by 2 (2 is a the factor of 15!+2)
15! +15 =15(14*13*12*11*10*9*8*7*6*5*4*3*2*1 + 1) > is completely divisible by 15 (15 is a the factor15!+15)
From here we can see that for any value of x in the range it's factor will only be less than x.
Therefore, we can definately say "yes" that n will be less than x. (Sufficient)



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Re: If x is an integer, does x have a factor n such that 1 <
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01 Jul 2018, 21:56
Bunuel wrote: zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) cannot be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B. Hey Bunuel, regarding statement 2: Let's say we would add a prime number bigger than 15 to the 15!, e.g. 17. Because it is not possible to factor out 17, the statement is not sufficient, right? Because the result might be prime or not. On the other hand, if we would add 16, we could still factor out 2 and 8 from 15! correct? Best regards chalmers15



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Re: If x is an integer, does x have a factor n such that 1 <
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01 Jul 2018, 23:58
chalmers15 wrote: Bunuel wrote: zisis wrote: If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15
I am sure i ve seen a quesiton like this one before, but tot forgot how to solve it............ If x is an integer, does x have a factor n such that 1 < n < x? Question basically asks: is \(x\) a prime number? If it is, then it won't have a factor \(n\) such that \(1<n<x\) (definition of a prime number). (1) \(x>3!\) > \(x\) is more than some number (3!). \(x\) may or may not be a prime. Not sufficient. (2) \(15!+2\leq{x}\leq{15!+15}\) > \(x\) cannot be a prime. For instance if \(x=15!+8=8*(2*3*4*5*6*7*9*10*11*12*13*14*15+1)\), then \(x\) is a multiple of 8, so not a prime. Same for all other numbers in this range: \(x=15!+k\), where \(2\leq{k}\leq{15}\) will definitely be a multiple of \(k\) (as weould be able to factor out \(k\) out of \(15!+k\)). Sufficient. Answer: B. Hey Bunuel, regarding statement 2: Let's say we would add a prime number bigger than 15 to the 15!, e.g. 17. Because it is not possible to factor out 17, the statement is not sufficient, right? Because the result might be prime or not. On the other hand, if we would add 16, we could still factor out 2 and 8 from 15! correct? Best regards chalmers15 You are almost right. Yes, 15! + 16 is not a prime because we can factor out an integer greater than 1 from 15! + 16. Yes, 15! + 17 might be a prime as well as not. But 15! + 17 is some specific number, so theoretically we could find out whether it's prime or not. If it were a prime, then \(15!+2\leq{x}\leq{15!+17}\) would not be sufficient and if it were NOT a prime, then \(15!+2\leq{x}\leq{15!+17}\) would be sufficient. Turns out that 15! + 17 is NOT a prime number: 15! + 17 = 1307674368017 = 9157×142805981. So, if (2) were \(15!+2\leq{x}\leq{15!+17}\) it would still be sufficient. But you won't get such number on the GMAT because you'll need a computer to check whether such a large number is a prime (there are some particular values of \(x=n!+1\) for which we can say whether it's a prime or not with help of Wilson's theorem, but again it's out of the scope of GMAT.).
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Re: If x is an integer, does x have a factor n such that 1 <
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19 Oct 2018, 08:23
VeritasPrepBrian wrote: I love this question, so I'll chime in even if only to bump it to the top so more people see it!
Hopefully most can see pretty quickly that x > 3! just means x > 6, and that isn't nearly enough to tell us whether it is prime.
Statement 2 is pretty neat, though: 15! + 2 ≤ x ≤ 15! + 15
Think about 15!. 15! is 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1, which means that:
15! is a multiple of 2 15! is a multiple of 3 15! is a multiple of 4 etc.... 15! is a multiple of 15
Now think about multiples of 2. Every SECOND number is a multiple of 2: 2, 4, 6, 8, 10. You create multiples of 2 by adding 2 to a previous multiple of 2. If a number is even, adding 2 "keeps it" even.
The same holds for 3. Every THIRD number (3, 6, 9, 12, 15, 18) is a multiple of 3. If you have a multiple of 3 and add 3 to it, it's still a multiple of 3.
This will hold for all of these numbers  because 15! is a multiple of every number between 2 and 15, then adding any number in that range of 215 will ensure that the new number remains a multiple of that number, and the new number will not be prime.
Therefore, statement 2 is sufficient, and the correct answer is B.
Now...consider the number 15! + 1. It's too big a number to know offhand whether it's prime, but we do know that it is NOT a multiple of any numbers 215. In order to be a multiple of 2, we'd have to add a multiple of 2 to 15!, and 1 breaks us off of that everysecondnumber cycle. Same for 3  we'd need to add a multiple of 3 in order to keep 15! on that everythird cycle, so 15! is not a multiple of 3. We can prove that 15! + 1 is not divisible by any numbers between 2 and 15. It's largest prime factor must then be 17 or greater.
Understanding that ideology and being able to determine divisibility of large numbers can be quite helpful on questions that might otherwise seem impossible. Thanks for posting this question! I have a question, if 15! + 1 is a prime number ? 15! + 1=1(15*14*13*.....*2+1) so 15! + 1 is a prime number?



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If x is an integer, does x have a factor n such that 1 <
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19 Oct 2018, 09:16
xiaoxueren wrote: I have a question, if 15! + 1 is a prime number ? 15! + 1=1(15*14*13*.....*2+1) so 15! + 1 is a prime number?
Yeah really good question. There *is* of course an answer to that, but not one you can prove in 23 minutes without a calculator so the GMAT couldn't really ask you about that. You *can* prove that it's not divisible by anything between 2 and 15, but you don't have any easy proof for prime numbers greater than 15 (could it be divisible by 17 or 31 or 53? We just don't know). Consider 5! + 1. 5! is 120 so 5! + 1 = 121. We know that 5! + 1 can't be divisible by 2, 3, 4, or 5, but the factorial setup doesn't give us any insight as to whether it's divisible by 11 (it is). So with 15! + 1, we can't predict (without a calculator at least) which larger prime factors it might have, so in the world of GMAT we can't prove whether it's prime or not.
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Re: If x is an integer, does x have a factor n such that 1 <
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19 Oct 2018, 21:42
VeritasPrepBrian wrote: I love this question, so I'll chime in even if only to bump it to the top so more people see it!
Hopefully most can see pretty quickly that x > 3! just means x > 6, and that isn't nearly enough to tell us whether it is prime.
Statement 2 is pretty neat, though: 15! + 2 ≤ x ≤ 15! + 15
Think about 15!. 15! is 15*14*13*12*11*10*9*8*7*6*5*4*3*2*1, which means that:
15! is a multiple of 2 15! is a multiple of 3 15! is a multiple of 4 etc.... 15! is a multiple of 15
Now think about multiples of 2. Every SECOND number is a multiple of 2: 2, 4, 6, 8, 10. You create multiples of 2 by adding 2 to a previous multiple of 2. If a number is even, adding 2 "keeps it" even.
The same holds for 3. Every THIRD number (3, 6, 9, 12, 15, 18) is a multiple of 3. If you have a multiple of 3 and add 3 to it, it's still a multiple of 3.
This will hold for all of these numbers  because 15! is a multiple of every number between 2 and 15, then adding any number in that range of 215 will ensure that the new number remains a multiple of that number, and the new number will not be prime.
Therefore, statement 2 is sufficient, and the correct answer is B.
Now...consider the number 15! + 1. It's too big a number to know offhand whether it's prime, but we do know that it is NOT a multiple of any numbers 215. In order to be a multiple of 2, we'd have to add a multiple of 2 to 15!, and 1 breaks us off of that everysecondnumber cycle. Same for 3  we'd need to add a multiple of 3 in order to keep 15! on that everythird cycle, so 15! is not a multiple of 3. We can prove that 15! + 1 is not divisible by any numbers between 2 and 15. It's largest prime factor must then be 17 or greater.
Understanding that ideology and being able to determine divisibility of large numbers can be quite helpful on questions that might otherwise seem impossible. Thanks for posting this question! Just loved it..Thanks. So this is how I applied it it for a small number. 4! + X here X can lie between 2 and 4 So, 4! + X will have factor from 4! to be keep it on the cycle of XIn short 4! + 2 cannot be prime since 4!+2 = 4*3*2*1+ 2=26 (26 factors as 2 ) Similarly 4! + 3 or 4!+ as long as the added term (except 1) is inside the factorial



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Re: If x is an integer, does x have a factor n such that 1 <
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22 Oct 2018, 04:59
VeritasPrepBrian wrote: xiaoxueren wrote: I have a question, if 15! + 1 is a prime number ? 15! + 1=1(15*14*13*.....*2+1) so 15! + 1 is a prime number?
Yeah really good question. There *is* of course an answer to that, but not one you can prove in 23 minutes without a calculator so the GMAT couldn't really ask you about that. You *can* prove that it's not divisible by anything between 2 and 15, but you don't have any easy proof for prime numbers greater than 15 (could it be divisible by 17 or 31 or 53? We just don't know). Consider 5! + 1. 5! is 120 so 5! + 1 = 121. We know that 5! + 1 can't be divisible by 2, 3, 4, or 5, but the factorial setup doesn't give us any insight as to whether it's divisible by 11 (it is). So with 15! + 1, we can't predict (without a calculator at least) which larger prime factors it might have, so in the world of GMAT we can't prove whether it's prime or not. Really appreciate, Now I understand this question.




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22 Oct 2018, 04:59






