It is currently 24 Feb 2018, 07:30

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# Integer x is equal to the product of all even numbers from 2 to 60, in

Author Message
TAGS:

### Hide Tags

Director
Joined: 25 Oct 2008
Posts: 591
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, 20:56
24
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

44% (01:35) correct 56% (01:46) wrong based on 256 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 x-1, 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
[Reveal] Spoiler: OA

_________________

http://gmatclub.com/forum/countdown-beginshas-ended-85483-40.html#p649902

SVP
Joined: 29 Aug 2007
Posts: 2467
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

13 Sep 2009, 22:33
1
KUDOS
1
This post was
BOOKMARKED
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 x-1, 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

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 (x-1):

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 (x-1), (x-1) must have a prime factore above 30.

Thats E.

Previous discussions are found here:

gmatprep-question-need-solution-74516.html#p556411
ps-function-74417.html#p556064
_________________

Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT

Manager
Joined: 22 Mar 2009
Posts: 77
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, 05:20
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 x-1, 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

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 (x-1):

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 (x-1), (x-1) must have a prime factore above 30.

Thats E.

Previous discussions are found here:

gmatprep-question-need-solution-74516.html#p556411
ps-function-74417.html#p556064

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 (x-1).I mean how can u be so sure that it wont be a factor..

kyle
_________________

Second cut is the deepest cut!!!:P

SVP
Joined: 29 Aug 2007
Posts: 2467
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

14 Sep 2009, 07:24
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 (x-1).I mean how can u be so sure that it wont be a factor..

kyle

Lets find y, a smallest prime factor of (x-1):

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.
_________________

Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT

Manager
Joined: 18 Jun 2010
Posts: 140
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

24 Oct 2011, 19:48
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, X-1 = 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 X-1 contains all prime numbers between 1 and 30, X-1 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: 32
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 06:15
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 x-1, 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, x-1 = 869 = 79x11 => y =11 ? smallest prime? so answer is C? no ?

Bunuel ? what do you think?
Math Expert
Joined: 02 Sep 2009
Posts: 43896
Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 06:46
4
KUDOS
Expert's post
12
This post was
BOOKMARKED
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 x-1, 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, x-1 = 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 $$x-1= 2^{30}*30!-1$$.

Now, two numbers $$x-1= 2^{30}*30!-1$$ and $$x=2^{30}*30!$$ are consecutive integers. Two consecutive integers are co-prime, 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 $$x-1= 2^{30}*30!-1$$ won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of $$x-1= 2^{30}*30!-1$$ must be more than 30.

Similar questions to practice:
for-every-positive-even-integer-n-the-function-h-n-is-126691.html
for-every-positive-even-integer-n-the-function-h-n-149722.html
x-is-the-product-of-all-even-numbers-from-2-to-50-inclusive-156545.html
if-n-is-a-positive-integer-greater-than-1-then-p-n-represe-144553.html
for-every-even-positive-integer-m-f-m-represents-the-produ-168636.html
for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html
if-a-and-b-are-odd-integers-a-b-represents-the-product-of-144714.html
the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html
for-every-positive-odd-integer-n-the-function-g-n-is-defined-to-be-t-181815.html
for-any-integer-n-greater-than-1-n-denotes-the-product-of-131701.html
let-p-be-the-product-of-the-positive-integers-between-1-and-132329.html
does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html
if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Hope it helps.
_________________
Manager
Joined: 10 Jun 2015
Posts: 126
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 07:40
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 x-1, 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

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 (x-1):

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 (x-1), (x-1) must have a prime factore above 30.

Thats E.

Previous discussions are found here:

gmatprep-question-need-solution-74516.html#p556411
ps-function-74417.html#p556064

If y were 29 then x-1 would be divisible by 29 however that is not the case is not proved
Manager
Joined: 10 Jun 2015
Posts: 126
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 07:49
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 x-1, 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, x-1 = 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 $$x-1= 2^{30}*30!-1$$.

Now, two numbers $$x-1= 2^{30}*30!-1$$ and $$x=2^{30}*30!$$ are consecutive integers. Two consecutive integers are co-prime, 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 $$x-1= 2^{30}*30!-1$$ won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of $$x-1= 2^{30}*30!-1$$ must be more than 30.

Similar questions to practice:
for-every-positive-even-integer-n-the-function-h-n-is-126691.html
for-every-positive-even-integer-n-the-function-h-n-149722.html
x-is-the-product-of-all-even-numbers-from-2-to-50-inclusive-156545.html
if-n-is-a-positive-integer-greater-than-1-then-p-n-represe-144553.html
for-every-even-positive-integer-m-f-m-represents-the-produ-168636.html
for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html
if-a-and-b-are-odd-integers-a-b-represents-the-product-of-144714.html
the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html
for-every-positive-odd-integer-n-the-function-g-n-is-defined-to-be-t-181815.html
for-any-integer-n-greater-than-1-n-denotes-the-product-of-131701.html
let-p-be-the-product-of-the-positive-integers-between-1-and-132329.html
does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html
if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Hope it helps.

If x=21 and x-1=20
factors of 21=1, 3, 7, 21; and 20 = 1, 2, 4, 5, 10, 20
x-1 has a factor which is less than 21 which is a factor of x
Math Expert
Joined: 02 Sep 2009
Posts: 43896
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 07:53
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 x-1, 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 $$x-1= 2^{30}*30!-1$$.

Now, two numbers $$x-1= 2^{30}*30!-1$$ and $$x=2^{30}*30!$$ are consecutive integers. Two consecutive integers are co-prime, 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 $$x-1= 2^{30}*30!-1$$ won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of $$x-1= 2^{30}*30!-1$$ must be more than 30.

Similar questions to practice:
for-every-positive-even-integer-n-the-function-h-n-is-126691.html
for-every-positive-even-integer-n-the-function-h-n-149722.html
x-is-the-product-of-all-even-numbers-from-2-to-50-inclusive-156545.html
if-n-is-a-positive-integer-greater-than-1-then-p-n-represe-144553.html
for-every-even-positive-integer-m-f-m-represents-the-produ-168636.html
for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html
if-a-and-b-are-odd-integers-a-b-represents-the-product-of-144714.html
the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html
for-every-positive-odd-integer-n-the-function-g-n-is-defined-to-be-t-181815.html
for-any-integer-n-greater-than-1-n-denotes-the-product-of-131701.html
let-p-be-the-product-of-the-positive-integers-between-1-and-132329.html
does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html
if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Hope it helps.

If x=21 and x-1=20
factors of 21=1, 3, 7, 21; and 20 = 1, 2, 4, 5, 10, 20
x-1 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: 32
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

20 Aug 2015, 09:13
1
This post was
BOOKMARKED
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 x-1, 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, x-1 = 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 $$x-1= 2^{30}*30!-1$$.

Now, two numbers $$x-1= 2^{30}*30!-1$$ and $$x=2^{30}*30!$$ are consecutive integers. Two consecutive integers are co-prime, 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 $$x-1= 2^{30}*30!-1$$ won't have ANY prime factor from 1 to 30. Hence y, the smallest prime factor of $$x-1= 2^{30}*30!-1$$ must be more than 30.

Similar questions to practice:
for-every-positive-even-integer-n-the-function-h-n-is-126691.html
for-every-positive-even-integer-n-the-function-h-n-149722.html
x-is-the-product-of-all-even-numbers-from-2-to-50-inclusive-156545.html
if-n-is-a-positive-integer-greater-than-1-then-p-n-represe-144553.html
for-every-even-positive-integer-m-f-m-represents-the-produ-168636.html
for-any-integer-p-p-is-equal-to-the-product-of-all-the-int-112494.html
if-a-and-b-are-odd-integers-a-b-represents-the-product-of-144714.html
the-function-f-m-is-defined-for-all-positive-integers-m-as-108309.html
for-every-positive-odd-integer-n-the-function-g-n-is-defined-to-be-t-181815.html
for-any-integer-n-greater-than-1-n-denotes-the-product-of-131701.html
let-p-be-the-product-of-the-positive-integers-between-1-and-132329.html
does-the-integer-k-have-a-factor-p-such-that-1-p-k-126735.html
if-x-is-an-integer-does-x-have-a-factor-n-such-that-100670.html

Hope it helps.

Many thanks Bunuel !!
Intern
Joined: 02 Jun 2015
Posts: 33
Location: United States
Concentration: Operations, Technology
GMAT Date: 08-22-2015
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, 12:55
Bunuel wrote:
Two consecutive integers are co-prime, 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: 01 Jun 2015
Posts: 3
GMAT 1: 720 Q50 V38
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

21 Aug 2015, 03:26
Here is how I approached this question -

x-1 = 2*4*6*8*10......*60

Let's take the example of 29 -
x-1 = 29*integer - 1 [hence not divisible by 29]

Similarly, for all the prime numbers < 30 ====> No Solution is possible.

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Note : Taking y=31, x-1 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: 5
Re: Integer x is equal to the product of all even numbers from 2 to 60, in [#permalink]

### Show Tags

28 Jan 2018, 08:15
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, x-1 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, x-1 CANNOT have 3 as a factor.

Continuing this way, we can conclude that x-1 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.

Re: Integer x is equal to the product of all even numbers from 2 to 60, in   [#permalink] 28 Jan 2018, 08:15
Display posts from previous: Sort by