If x, y, and z are consecutive prime numbers and x < y < z, which of the following must be true?
I. (x+y)/2 is an integer.
II. xyz has exactly 8 factors.
III. y - x = 1
a) I only
b) II only
c) III only
d) I and II only
e) II, and III
prime numbers
This topic has expert replies
-
- Junior | Next Rank: 30 Posts
- Posts: 25
- Joined: Wed Nov 17, 2010 7:54 am
- Thanked: 1 times
GMAT/MBA Expert
- Anju@Gurome
- GMAT Instructor
- Posts: 511
- Joined: Wed Aug 11, 2010 9:47 am
- Location: Delhi, India
- Thanked: 344 times
- Followed by:86 members
As x, y, and z are prime, number of factors of xyz is (1 + 1)(1 + 1)(1 + 1) = 8vishal_2804 wrote:If x, y, and z are consecutive prime numbers and x < y < z, which of the following must be true?
I. (x+y)/2 is an integer.
II. xyz has exactly 8 factors.
III. y - x = 1
So, II must be true.
Consider x = 2, y = 3, and z = 5
- I. (x + y)/2 = (2 + 3)/2 = 2.5 ---> Not an integer ---> I may not be true
- I. y - x = (5 - 3) = 2 ---> III may not be true
Anju Agarwal
Quant Expert, Gurome
Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.
§ GMAT with Gurome § Admissions with Gurome § Career Advising with Gurome §
Quant Expert, Gurome
Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.
§ GMAT with Gurome § Admissions with Gurome § Career Advising with Gurome §
-
- Junior | Next Rank: 30 Posts
- Posts: 25
- Joined: Wed Nov 17, 2010 7:54 am
- Thanked: 1 times
can u plz explain the logic for As x, y, and z are prime, number of factors of xyz is (1 + 1)(1 + 1)(1 + 1) = 8.
Thanks
Thanks
GMAT/MBA Expert
- Anju@Gurome
- GMAT Instructor
- Posts: 511
- Joined: Wed Aug 11, 2010 9:47 am
- Location: Delhi, India
- Thanked: 344 times
- Followed by:86 members
If the prime factorization of a positive integer N is (p^a)*(q^b)*(r^c)..., the number of different positive factors of N is given by (a + 1)(b + 1)(c + 1)...vishal_2804 wrote:can u plz explain the logic for As x, y, and z are prime, number of factors of xyz is (1 + 1)(1 + 1)(1 + 1) = 8.
Thanks
This is because any factor of N,
- may contain either p� or p¹ or p² ... or pᵃ ---> (a + 1) ways to select or not select p
may contain either q� or q¹ or q² ... or qᵇ ---> (b + 1) ways to select or not select q
may contain either r� or r¹ or r² ... or rᶜ ---> (c + 1) ways to select or not select r
Ans so on...
Hence, the steps for finding out the number of different positive factors of any integer N are...
- 1. Find out the prime factorization of N
2. Increment the powers of all the prime factors by 1.
3. Multiply the incremented powers.
- Number of different positive factors of 12 = (2²)*(3¹) is (2 + 1)*(1 + 1) = 6
Number of different positive factors of 360 = (2³)*(3²)*(5¹) is (3 + 1)*(2 + 1)*(1 + 1) = 24
So, number of different positive factors of xyz is (1 + 1)*(1 + 1)*(1 + 1) = 8
Hope that helps.
Anju Agarwal
Quant Expert, Gurome
Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.
§ GMAT with Gurome § Admissions with Gurome § Career Advising with Gurome §
Quant Expert, Gurome
Backup Methods : General guide on plugging, estimation etc.
Wavy Curve Method : Solving complex inequalities in a matter of seconds.
§ GMAT with Gurome § Admissions with Gurome § Career Advising with Gurome §
-
- Senior | Next Rank: 100 Posts
- Posts: 46
- Joined: Sun Sep 12, 2010 12:32 pm
- Thanked: 1 times
- Followed by:1 members
Thanks for the response Anju ... does this formula hold even with multiplication of prime numbers with a common factor.
E.g. 13 , 17 , 19 ... each of the primes have only 2 factors (13 & 1 / 17 & 1 / 19 &1) whereas the product should only have 4 factors and not 8 factors.. not sure what am I missing oh 13*17 is also a factor .. so that adds 2 more factors .. so I guess since we have a common factor we aren't we counting it multiple times ?
Also the optimus prime 2 was a good.. 1) holds true for every other prime also holds true for (y + z)/2..
E.g. 13 , 17 , 19 ... each of the primes have only 2 factors (13 & 1 / 17 & 1 / 19 &1) whereas the product should only have 4 factors and not 8 factors.. not sure what am I missing oh 13*17 is also a factor .. so that adds 2 more factors .. so I guess since we have a common factor we aren't we counting it multiple times ?
Also the optimus prime 2 was a good.. 1) holds true for every other prime also holds true for (y + z)/2..