Author 
Message 
TAGS:

Hide Tags

Senior Manager
Joined: 25 Oct 2008
Posts: 438
Location: Kolkata,India

Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
13 Sep 2009, 21:56
Question Stats:
49% (02:20) correct 51% (02:21) wrong based on 213 sessions
HideShow timer Statistics
Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true? (A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30
Official Answer and Stats are available only to registered users. Register/ Login.




Math Expert
Joined: 02 Sep 2009
Posts: 60490

Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 07:46
jimwild wrote: tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true?
(A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30 so the sum is 29 and x = 870, x1 = 869 = 79x11 => y =11 ? smallest prime? so answer is C? no ? Bunuel ? what do you think? x is equal to the product of all even numbers from 2 to 60, inclusive: \(x = 2*4*6*...*58*60=(2*1)*(2*2)*(2*3)*...*(2*29)*(2*30)=2^{30}*30!\). y is the smallest prime number that is also a factor of \(x1= 2^{30}*30!1\). Now, two numbers \(x1= 2^{30}*30!1\) and \(x=2^{30}*30!\) are consecutive integers. Two consecutive integers are coprime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. Since \(x=2^{30}*30!\) has all prime numbers from 1 to 30 as its factors, then according to the above \(x1= 2^{30}*30!1\) won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of \(x1= 2^{30}*30!1\) must be more than 30. Answer: E. Similar questions to practice: foreverypositiveevenintegernthefunctionhnis126691.htmlforeverypositiveevenintegernthefunctionhn149722.htmlxistheproductofallevennumbersfrom2to50inclusive156545.htmlifnisapositiveintegergreaterthan1thenpnreprese144553.htmlforeveryevenpositiveintegermfmrepresentstheprodu168636.htmlforanyintegerppisequaltotheproductofalltheint112494.htmlifaandbareoddintegersabrepresentstheproductof144714.htmlthefunctionfmisdefinedforallpositiveintegersmas108309.htmlforeverypositiveoddintegernthefunctiongnisdefinedtobet181815.htmlforanyintegerngreaterthan1ndenotestheproductof131701.htmlletpbetheproductofthepositiveintegersbetween1and132329.htmldoestheintegerkhaveafactorpsuchthat1pk126735.htmlifxisanintegerdoesxhaveafactornsuchthat100670.htmlHope it helps.
_________________




SVP
Joined: 29 Aug 2007
Posts: 1781

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
13 Sep 2009, 23:33
tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60. If y is the smallest prime no. that is also a factor of x1, then which of the following expressions must be true?
y lies bet. 0 and 4 y lies bet. 4 and 10 y lies bet. 10 and 20 y lies bet. 20 and 30 y is greater than 30
Guys please help me understand the underlying principles.. This is a replication of one of the most discussed and hardest question (I guess the question is from retired gmat question bank). Per the question, x = the product of all even numbers from 2 to n, where n is any even integer value from 2 to 60 including. If x = 2, x  1 = 2! 1 If x = 2x4, x  1 = 2! (2^2)  1 If x = 2x4x6, x  1 = 3! (2^3)  1 If x = 2x4x6x8, x  1 = 4! (2^4)  1 . . . . If x = 2x4x6x.......x60, x  1 = 30! (2^30)  1 Lets find y, a smallest prime factor of (x1): If y were 2, [(30!) (2^30) (1)] would be divisible by 2 however thats not the case. If y were 3, [(30!) (2^30) (1)] would be divisible by 3 however thats not the case. If y were 5, [(30!) (2^30) (1)] would be divisible by 5 however thats not the case. . . . . . If y were 29, [(30!) (2^30) (1)] would be divisible by 29 however thats not the case. If none of the primes under 31 is a factor of (x1), (x1) must have a prime factore above 30. Thats E. Previous discussions are found here: gmatprepquestionneedsolution74516.html#p556411 psfunction74417.html#p556064



Manager
Joined: 22 Mar 2009
Posts: 56
Schools: Darden:Tepper:UCUIC:Kenan Flager:Nanyang:NUS:ISB:UCI Merage:Emory
WE 1: 3

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
14 Sep 2009, 06:20
GMAT TIGER wrote: tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60. If y is the smallest prime no. that is also a factor of x1, then which of the following expressions must be true?
y lies bet. 0 and 4 y lies bet. 4 and 10 y lies bet. 10 and 20 y lies bet. 20 and 30 y is greater than 30
Guys please help me understand the underlying principles.. This is a replication of one of the most discussed and hardest question (I guess the question is from retired gmat question bank). Per the question, x = the product of all even numbers from 2 to n, where n is any even integer value from 2 to 60 including. If x = 2, x  1 = 2! 1 If x = 2x4, x  1 = 2! (2^2)  1 If x = 2x4x6, x  1 = 3! (2^3)  1 If x = 2x4x6x8, x  1 = 4! (2^4)  1 . . . . If x = 2x4x6x.......x60, x  1 = 30! (2^30)  1 Lets find y, a smallest prime factor of (x1): If y were 2, [(30!) (2^30) (1)] would be divisible by 2 however thats not the case. If y were 3, [(30!) (2^30) (1)] would be divisible by 3 however thats not the case. If y were 5, [(30!) (2^30) (1)] would be divisible by 5 however thats not the case. . . . . . If y were 29, [(30!) (2^30) (1)] would be divisible by 29 however thats not the case. If none of the primes under 31 is a factor of (x1), (x1) must have a prime factore above 30. Thats E. Previous discussions are found here: gmatprepquestionneedsolution74516.html#p556411 psfunction74417.html#p556064dear gmatiger, i was able to understand the x part...but hw can u say the y=2 , 3 ,5 etc etc be not a factor of (x1).I mean how can u be so sure that it wont be a factor.. please explain in brief. thanks in advance , kyle



SVP
Joined: 29 Aug 2007
Posts: 1781

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
14 Sep 2009, 08:24
kylexy wrote: dear gmatiger,
i was able to understand the x part...but hw can u say the y=2 , 3 ,5 etc etc be not a factor of (x1).I mean how can u be so sure that it wont be a factor..
please explain in brief. thanks in advance , kyle Lets find y, a smallest prime factor of (x1): If y were 2, [30! (2^30)  1]/2 must be divisible by 2 and result in an integer. Lets do little work on it: = [30! (2^30) 1] / 2 = [30! (2^30)/2  1/2] = [15 x 29! (2^30)  1/2] ......... is it an integer? No. If it were, then 2 would be a smallest prime. Similarly: = [30! (2^30)  1] / 3 = [30! (2^30) / 3  1/3] = [10 x 29! (2^30)  1/3] ......... is it an integer? No. If it were, then 3 would be a smallest prime. Also = [30! (2^30)  1] / 29 = [30! (2^30)/29  1/29] = [30 x 28! (2^30)  1/29] ......... is it an integer? No. If it were, then 3 would be a smallest prime. So what would be y, the smallest prime factor? It must be greater than 29 i.e. greater than 30 too. Hope it is clear now.



Manager
Joined: 18 Jun 2010
Posts: 81

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
24 Oct 2011, 20:48
X can be written as = 2.4.6.8....60
or X = 2.(2.2)(2.3)(2.4)....(2.30) i.e. X = 2^30(1.2.3.4.5...30)
so, X1 = 2^30(1.2.3....30)1
Now take a small example, prime factorization of a number 10 = 1*2*5, if i add or subtract 1 from the number then the prime factors involved (i.e. 2 and 5) will no longer divide 9 or 11.
Going by the same principle, if X1 contains all prime numbers between 1 and 30, X1 is bound to be divisible by a prime number that is not already present in X.
Greater than 30 is the only possible choice. Hence E.



Intern
Joined: 02 Mar 2015
Posts: 29

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 07:15
tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true?
(A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30 so the sum is 29 and x = 870, x1 = 869 = 79x11 => y =11 ? smallest prime? so answer is C? no ? Bunuel ? what do you think?



Manager
Joined: 10 Jun 2015
Posts: 110

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 08:40
GMAT TIGER wrote: tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60. If y is the smallest prime no. that is also a factor of x1, then which of the following expressions must be true?
y lies bet. 0 and 4 y lies bet. 4 and 10 y lies bet. 10 and 20 y lies bet. 20 and 30 y is greater than 30
Guys please help me understand the underlying principles.. This is a replication of one of the most discussed and hardest question (I guess the question is from retired gmat question bank). Per the question, x = the product of all even numbers from 2 to n, where n is any even integer value from 2 to 60 including. If x = 2, x  1 = 2! 1 If x = 2x4, x  1 = 2! (2^2)  1 If x = 2x4x6, x  1 = 3! (2^3)  1 If x = 2x4x6x8, x  1 = 4! (2^4)  1 . . . . If x = 2x4x6x.......x60, x  1 = 30! (2^30)  1 Lets find y, a smallest prime factor of (x1): If y were 2, [(30!) (2^30) (1)] would be divisible by 2 however thats not the case. If y were 3, [(30!) (2^30) (1)] would be divisible by 3 however thats not the case. If y were 5, [(30!) (2^30) (1)] would be divisible by 5 however thats not the case. . . . . . If y were 29, [(30!) (2^30) (1)] would be divisible by 29 however thats not the case. If none of the primes under 31 is a factor of (x1), (x1) must have a prime factore above 30. Thats E. Previous discussions are found here: gmatprepquestionneedsolution74516.html#p556411 psfunction74417.html#p556064If y were 29 then x1 would be divisible by 29 however that is not the case is not proved



Manager
Joined: 10 Jun 2015
Posts: 110

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 08:49
Bunuel wrote: jimwild wrote: tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true?
(A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30 so the sum is 29 and x = 870, x1 = 869 = 79x11 => y =11 ? smallest prime? so answer is C? no ? Bunuel ? what do you think? x is equal to the product of all even numbers from 2 to 60, inclusive: \(x = 2*4*6*...*58*60=(2*1)*(2*2)*(2*3)*...*(2*29)*(2*30)=2^{30}*30!\). y is the smallest prime number that is also a factor of \(x1= 2^{30}*30!1\). Now, two numbers \(x1= 2^{30}*30!1\) and \(x=2^{30}*30!\) are consecutive integers. Two consecutive integers are coprime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. Since \(x=2^{30}*30!\) has all prime numbers from 1 to 30 as its factors, then according to the above \(x1= 2^{30}*30!1\) won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of \(x1= 2^{30}*30!1\) must be more than 30. Answer: E. Similar questions to practice: foreverypositiveevenintegernthefunctionhnis126691.htmlforeverypositiveevenintegernthefunctionhn149722.htmlxistheproductofallevennumbersfrom2to50inclusive156545.htmlifnisapositiveintegergreaterthan1thenpnreprese144553.htmlforeveryevenpositiveintegermfmrepresentstheprodu168636.htmlforanyintegerppisequaltotheproductofalltheint112494.htmlifaandbareoddintegersabrepresentstheproductof144714.htmlthefunctionfmisdefinedforallpositiveintegersmas108309.htmlforeverypositiveoddintegernthefunctiongnisdefinedtobet181815.htmlforanyintegerngreaterthan1ndenotestheproductof131701.htmlletpbetheproductofthepositiveintegersbetween1and132329.htmldoestheintegerkhaveafactorpsuchthat1pk126735.htmlifxisanintegerdoesxhaveafactornsuchthat100670.htmlHope it helps. If x=21 and x1=20 factors of 21=1, 3, 7, 21; and 20 = 1, 2, 4, 5, 10, 20 x1 has a factor which is less than 21 which is a factor of x



Math Expert
Joined: 02 Sep 2009
Posts: 60490

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 08:53
matvan wrote: Bunuel wrote: jimwild wrote: Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true? (A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30 x is equal to the product of all even numbers from 2 to 60, inclusive: \(x = 2*4*6*...*58*60=(2*1)*(2*2)*(2*3)*...*(2*29)*(2*30)=2^{30}*30!\). y is the smallest prime number that is also a factor of \(x1= 2^{30}*30!1\). Now, two numbers \(x1= 2^{30}*30!1\) and \(x=2^{30}*30!\) are consecutive integers. Two consecutive integers are coprime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. Since \(x=2^{30}*30!\) has all prime numbers from 1 to 30 as its factors, then according to the above \(x1= 2^{30}*30!1\) won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of \(x1= 2^{30}*30!1\) must be more than 30. Answer: E. Similar questions to practice: foreverypositiveevenintegernthefunctionhnis126691.htmlforeverypositiveevenintegernthefunctionhn149722.htmlxistheproductofallevennumbersfrom2to50inclusive156545.htmlifnisapositiveintegergreaterthan1thenpnreprese144553.htmlforeveryevenpositiveintegermfmrepresentstheprodu168636.htmlforanyintegerppisequaltotheproductofalltheint112494.htmlifaandbareoddintegersabrepresentstheproductof144714.htmlthefunctionfmisdefinedforallpositiveintegersmas108309.htmlforeverypositiveoddintegernthefunctiongnisdefinedtobet181815.htmlforanyintegerngreaterthan1ndenotestheproductof131701.htmlletpbetheproductofthepositiveintegersbetween1and132329.htmldoestheintegerkhaveafactorpsuchthat1pk126735.htmlifxisanintegerdoesxhaveafactornsuchthat100670.htmlHope it helps. If x=21 and x1=20 factors of 21=1, 3, 7, 21; and 20 = 1, 2, 4, 5, 10, 20 x1 has a factor which is less than 21 which is a factor of x The question talks about prime factors, not just factors.
_________________



Intern
Joined: 02 Mar 2015
Posts: 29

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 10:13
Bunuel wrote: jimwild wrote: tejal777 wrote: Integer x is equal to the product of all even numbers from 2 to 60, inclusive. If y is the smallest prime number that is also a factor of x1, then which of the following expressions must be true?
(A) 0<y<4 (B) 4<y<10 (C) 10<y<20 (D) 20<y<30 (E) y>30 so the sum is 29 and x = 870, x1 = 869 = 79x11 => y =11 ? smallest prime? so answer is C? no ? Bunuel ? what do you think? x is equal to the product of all even numbers from 2 to 60, inclusive: \(x = 2*4*6*...*58*60=(2*1)*(2*2)*(2*3)*...*(2*29)*(2*30)=2^{30}*30!\). y is the smallest prime number that is also a factor of \(x1= 2^{30}*30!1\). Now, two numbers \(x1= 2^{30}*30!1\) and \(x=2^{30}*30!\) are consecutive integers. Two consecutive integers are coprime, which means that they don't share ANY common factor but 1. For example 20 and 21 are consecutive integers, thus only common factor they share is 1. Since \(x=2^{30}*30!\) has all prime numbers from 1 to 30 as its factors, then according to the above \(x1= 2^{30}*30!1\) won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of \(x1= 2^{30}*30!1\) must be more than 30. Answer: E. Similar questions to practice: foreverypositiveevenintegernthefunctionhnis126691.htmlforeverypositiveevenintegernthefunctionhn149722.htmlxistheproductofallevennumbersfrom2to50inclusive156545.htmlifnisapositiveintegergreaterthan1thenpnreprese144553.htmlforeveryevenpositiveintegermfmrepresentstheprodu168636.htmlforanyintegerppisequaltotheproductofalltheint112494.htmlifaandbareoddintegersabrepresentstheproductof144714.htmlthefunctionfmisdefinedforallpositiveintegersmas108309.htmlforeverypositiveoddintegernthefunctiongnisdefinedtobet181815.htmlforanyintegerngreaterthan1ndenotestheproductof131701.htmlletpbetheproductofthepositiveintegersbetween1and132329.htmldoestheintegerkhaveafactorpsuchthat1pk126735.htmlifxisanintegerdoesxhaveafactornsuchthat100670.htmlHope it helps. Many thanks Bunuel !!



Intern
Joined: 02 Jun 2015
Posts: 32
Location: United States
Concentration: Operations, Technology
Schools: HBS '18, Stanford '18, Wharton '18, Kellogg '18, Booth '18, Sloan '18, Ross '18, Haas '18, Tuck '18, Yale '18, Duke '18, Anderson '18, Darden '18, Tepper '18, Marshall '18, UFlorida '18
GMAT Date: 08222015
GPA: 3.92
WE: Science (Other)

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
20 Aug 2015, 13:55
Bunuel wrote: Two consecutive integers are coprime, which means that they don't share ANY common factor but 1. This is a beautiful and quick way to solve the problem. Thanks. =)



Intern
Status: Time to Improvize
Joined: 02 Jun 2015
Posts: 3

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
21 Aug 2015, 04:26
Here is how I approached this question 
x1 = 2*4*6*8*10......*60
Let's take the example of 29  x1 = 29*integer  1 [hence not divisible by 29]
Similarly, for all the prime numbers < 30 ====> No Solution is possible.
Answer = E  Note : Taking y=31, x1 will NOT have 31 or any of it's multiple 62,93, etc.. hence, it still has a chance of being a factor.



Intern
Joined: 08 Jul 2017
Posts: 20

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
28 Jan 2018, 09:15
x= 2x4x.....x60 = (2x1)x(2x2)x.....x(2x30) = (2^30)x(1x2x...30) = (2^30)x(30!) Clearly, as 30! is a factor of x, so is every prime number less than or equal to 30. Hence, 2, 3, 5,....,29 are all factors of x. Consider 2. We know that among consecutive integers, every second integer has 2 as a factor (since every second integer is even). This means that if x has 2 as a factor, x1 CANNOT have 2 as a factor. (since they're consecutive, they cannot both be even) Consider 3.We know that among consecutive integers, every third integer has 3 as a factor. This means that if x has 3 as a factor, x1 CANNOT have 3 as a factor. Continuing this way, we can conclude that x1 would not have any of the prime factors till 29. Thus, the only prime factors it will have have to be greater than 29. Since 30 is not prime, we can say that the only prime factors it will have have to be greater than 30. Hence, the answer is E.
_________________



NonHuman User
Joined: 09 Sep 2013
Posts: 13986

Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
Show Tags
15 Jun 2019, 23:44
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________




Re: Integer x is equal to the product of all even numbers from 2 to 60, in
[#permalink]
15 Jun 2019, 23:44






