Help on a PS question please.
Help on a PS question please.
This topic has expert replies
-
- Junior | Next Rank: 30 Posts
- Posts: 18
- Joined: Sat May 04, 2013 9:49 am
- theCodeToGMAT
- Legendary Member
- Posts: 1556
- Joined: Tue Aug 14, 2012 11:18 pm
- Thanked: 448 times
- Followed by:34 members
- GMAT Score:650
h(100) = 2 x 4 x 6 x 8 ..... 100
h(100) = 2(1x2x3..50) = (2^50)*50!
h(100) + 1 = (2^50)*50! + 1
here, 2 to 50 are factors of h(100) so they cannot be factor of h(100)+1
So, the factor must be greater than 50.
Answer [spoiler]{E}[/spoiler]
h(100) = 2(1x2x3..50) = (2^50)*50!
h(100) + 1 = (2^50)*50! + 1
here, 2 to 50 are factors of h(100) so they cannot be factor of h(100)+1
So, the factor must be greater than 50.
Answer [spoiler]{E}[/spoiler]
R A H U L
-
- Junior | Next Rank: 30 Posts
- Posts: 18
- Joined: Sat May 04, 2013 9:49 am
Thanks Rahul for the explanation.
Can you please elaborate on the below point?
here, 2 to 50 are factors of h(100) so they cannot be factor of h(100)+1
Can you please elaborate on the below point?
here, 2 to 50 are factors of h(100) so they cannot be factor of h(100)+1
GMAT/MBA Expert
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Important Concept: If k is a positive integer that's greater than 1, and if k is a factor (divisor) of N, then k is not a divisor of N+1For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, the p is
A: Between 2 & 10
B: Between 10 & 20
C: Between 20 & 30
D: Between 30 & 40
E: Greater than 40
For example, since 7 is a factor of 350, we know that 7 is not a factor of (350+1)
Similarly, since 8 is a factor of 312, we know that 8 is not a factor of 313
Now let's examine h(100)
h(100) = (2)(4)(6)(8)....(96)(98)(100)
= (2x1)(2x2)(2x3)(2x4)....(2x48)(2x49)(2x50)
Factor out all of the 2's to get: h(100) = [2^50][(1)(2)(3)(4)....(48)(49)(50)]
Since 2 is in the product of h(100), we know that 2 is a factor of h(100), which means that 2 is not a factor of h(100)+1 (based on the above rule)
Similarly, since 3 is in the product of h(100), we know that 3 is a factor of h(100), which means that 3 is not a factor of h(100)+1 (based on the above rule)
Similarly, since 5 is in the product of h(100), we know that 5 is a factor of h(100), which means that 5 is not a factor of h(100)+1 (based on the above rule)
.
.
.
.
Similarly, since 47 is in the product of h(100), we know that 47 is a factor of h(100), which means that 47 is not a factor of h(100)+1 (based on the above rule)
So, we can see that none of the primes from 2 to 47 can be factors of h(100)+1, which means the smallest prime factor of h(100)+1 must be greater than 47.
Answer = E
Cheers,
- theCodeToGMAT
- Legendary Member
- Posts: 1556
- Joined: Tue Aug 14, 2012 11:18 pm
- Thanked: 448 times
- Followed by:34 members
- GMAT Score:650
I will give you examples to explain this.srinivasapriyan.r wrote:Thanks Rahul for the explanation.
Can you please elaborate on the below point?
here, 2 to 50 are factors of h(100) so they cannot be factor of h(100)+1
Consider 15 = 3 x 5 .. Now if we increase 15 by 1 i.e. 16 = 2x2x2x2 --> both "3" & "5" are not factors
Similarly 20 = 2x5x2 .. & 21 = 7x3 --> No common factors
Similarly 50 = 2x5x5 ... & 51 = 3x17 ---> No common factors
So, if a number has some factors . .then number+1 will not have same factors..
Same logic implies here.
I hope it's better now
R A H U L
- GMATGuruNY
- GMAT Instructor
- Posts: 15539
- Joined: Tue May 25, 2010 12:04 pm
- Location: New York, NY
- Thanked: 13060 times
- Followed by:1906 members
- GMAT Score:790
Since the difference between them is 1, h(100) and h(100)+1 are consecutive integers.For every positive even integer n, the function h(n) is defined to be the product of all even integers from 2 to n, inclusive. If p is the smallest prime factor of h(100) + 1, the p is
A: Between 2 & 10
B: Between 10 & 20
C: Between 20 & 30
D: Between 30 & 40
E: Greater than 40
Consecutive integers are COPRIMES: they share no factors other than 1.
Let's examine why:
If x is a multiple of 2, the next largest multiple of 2 is x+2.
If x is a multiple of 3, the next largest multiple of 3 is x+3.
Using this logic, if we go from x to x+1, we get only to the next largest multiple of 1.
So 1 is the only factor common both to x and to x+1.
In other words, x and x+1 are COPRIMES.
Thus:
h(100) and h(100)+1 are COPRIMES. They share no factors other than 1.
h(100) = 2 * 4 * 6 *....* 94 * 96 * 98 * 100
Factoring out 2 from every value above, we get:
h(100) = 2��(1 * 2 * 3 *... * 47 * 48 * 49 * 50)
Looking at the set of parentheses on the right, we can see that every prime number between 1 and 50 is a factor of h(100).
Since h(100) and h(100)+1 are coprimes, NONE of the prime numbers between 1 and 50 can be a factor of h(100)+1.
Thus, the smallest prime factor of h(100) + 1 must be greater than 50.
The correct answer is E.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at [email protected].
Student Review #1
Student Review #2
Student Review #3