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# If x, y, and z are positive integers such that x < y < z

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Manager
Joined: 04 Nov 2006
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Location: California
If x, y, and z are positive integers such that x < y < z  [#permalink]

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20 Dec 2006, 16:05
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55
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95% (hard)

Question Stats:

35% (02:36) correct 65% (02:39) wrong based on 827 sessions

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If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57
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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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27 May 2012, 04:57
20
12
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

Notice that we are told that z is an odd number.

(1) x and y are prime numbers, whose sum is a factor of 57 --> factors of 57 are 1, 3, 19 and 57. Since we are told that x and y are prime numbers then x can only be 2 and y can only be 17 (x+y=19). x=2=even cannot be a factor of the odd integer z. Sufficient.

(2) z is a factor of 57. Clearly insufficient as we have no info about x.

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Re: DS_If x, y, and z are positive integers...  [#permalink]

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27 May 2012, 01:55
6
Swagatalakshmi wrote:
mm007 wrote:
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57

57 has following factors 1,3, 19, 57

From (1) :

x+y =3 if x =1 then x is not prime
x+y =19 Therefore x=2, y=17
x+y =57 can't be expressed as sum of two prime numbers.

Answer is A. Since x has to be 2, and it is given in the question that z is an odd integer.
Thus we know that 2 cannot be the factor of an odd integer.

##### General Discussion
Manager
Joined: 04 Jan 2006
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20 Dec 2006, 16:50
From stmt 1:
factors of 57 are 1,3,19,57
For x and y to be prome numbers and their sum to be any of these
x has to be 2 and y has to 17.
We don't know anything about z.
INSUFF

From stmt 2:
Z is 1 or 3 or 19 or 57.
INSUFF

Combining:
Z can not be 1 or 3.(x and y are prime and they are less than z)
2 is not a factor of 19 or 57.

So C.

One question though,
It is time consuming to find whether 57 is the sum of any prime numbers.
I had to pick the primes below 57 and find out whether 2 primes can sum up to 57.
Is there any easy way?
Director
Joined: 30 Nov 2006
Posts: 546
Location: Kuwait

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20 Dec 2006, 17:03

Here is the reason:

Question gives: x < y < z, each is +int
z is odd +int

Question asks: is X factor of Z ? [ does X*K = Z, K:+int ]

Statement 1: X & Y are prime and X+Y factor of 57
-----------------------------------------------------------
57 is a prime number --> x+y =1 or 57
since x and y are each +tive int., then neither x nor y can be zero
Therefore, x+y = 57

Note: you don't really need to find what X and Y are

Statement 2: Z is a factor of 57
--------------------------------------
57 is prime, Z can not be 1 because z>y>z and non of x and y can be zero or negative
So Z = 57
Yet, this tells us nothing about x and y

Statements 1 and 2
-----------------------
from 2, z = 57 and the only factors of 57 are 1 and 57
from question, neither x nor y can be zero or 1
from statement 1, neither x nor y can be 57
So, X is NOT a factor of Z

Is it helpful, the way I expain ? or is too much details and too confusing ?
Senior Manager
Joined: 23 Jun 2006
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16 Jan 2007, 14:47
3
there is an easy way to verify that 57 is not the sum of primes...
for a sum of two numbers to be odd - one must be even and one odd. if these are primes then the even number must be 2 and the odd must be 55 which is not a prime ... hence no solution to that. no need to pick numbers.

amit.
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Joined: 20 Feb 2006
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16 Jan 2007, 15:55
dr shpak - it doesn't say they are consecutive numbers.

My initial reaction to this was E and I agree with Hobbit.
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16 Jan 2007, 16:04
exactly , no way sum of two primes can equate 57. if x + y = 57, then one of them has to be even, and the only even prime is 2.
==> y has to be 55 which is not prime.

so E.
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Re: DS_If x, y, and z are positive integers...  [#permalink]

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17 Jan 2007, 00:03
mm007 wrote:
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57

57 has following factors 1,3, 19, 57

From (1) :

x+y =3 if x =1 then x is not prime
x+y =19 Therefore x=2, y=17
x+y =57 can't be expressed as sum of two prime numbers.

From (2) :
z can be 3,19 or 57 ...... INSUFF

Combining (1) and (2), we get :

when x=2, y=17 and z=19 or 57, x is not a factor of z

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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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09 Feb 2014, 05:53
2
1

Z is odd from question stem.

statement 1 says x & y are primes whose sum is a factor of 57. First lets deduce factors of 57.

57 = 1 * 57
= 3 * 19

so x+y must be equal to one of these factors. x+y cannot be equal to 1 or 3 since x & y are positive prime integers.

we have x+y=57 or x+y = 19.
Lets take x+y=57 for this to happen one of the x & y must be even and remember both must be prime so only even prime is 2. if x=2 then y will be 55 and not prime so x+y = 57 is ruled out.

taking what is left. x+y=19 and using same logic as above x has to be 2. and y=17 hence both are prime and sum is factor of 57. Going back to the question, is x a factor of odd integer z? NO x=2 it cannot be. Sufficient!

Statement 2 is clearly insufficient. no information about x.
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Re: If x, y, and z are positive integers such that x < y < z, is x a facto  [#permalink]

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12 Nov 2014, 00:55
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57

Statement 1:
Factors of 57 = 3 * 19
Therefore X, Y, Z could be:
1, 2, 3 >> Yes
2, 17, 19 >> No
INSUFFICIENT

Statement 2:
z is a factor of 57, meaning z is either 3 or 19
However, we don't know anything about x
INSUFFICIENT

Statements 1+2:
If x+y is 3 or 19
and
z is 3 or 19
The numbers could be:
1, 18, 19 >> Yes (because 1 is a factor of 19)
1, 2, 3 >> Yes (because 1 is a factor of 3)
8, 11, 19 >> No
INSUFFICIENT

Where am I going wrong here? How is the answer A?
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Re: If x, y, and z are positive integers such that x < y < z, is x a facto  [#permalink]

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12 Nov 2014, 01:05
kthxbye wrote:
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57

Statement 1:
Factors of 57 = 3 * 19
Therefore X, Y, Z could be:
1, 2, 3 >> Yes
2, 17, 19 >> No
INSUFFICIENT

Statement 1 also says that x and y are prime numbers. 1 is not a prime. It's always a No with statement 1 and hence is sufficient. Will post detailed solution tonight.

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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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12 Nov 2014, 04:18
Hi,

Statement 1: x and y are prime numbers, whose sum is a factor of 57

57= 1 x 57 &
57= 3 x 19
Factors are 1, 3, 19, 57
1, 3 and 57 cannot be the sum!!
( 3 = 2+1 but 1 is not a prime number; try any combination for 57, it will always be the sum of an even integer and odd integer and 2 is the only even prime number. So you cannot have 57 as the sum of two prime numbers!!!)

So the sum should be 19
and 19 = 2 + 17 is the only way you can write it as a sum of two prime numbers.

So the numbers are x,y, z which is 2,17, an odd integer!!
Now ask the question : Is x a factor of odd integer z?
The answer is No for sure. Hence Statement 1 alone is sufficient!!

Statement 2 alone doesn't tell you anything about x, so you cannot say that x is a factor of 57!!

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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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23 May 2015, 00:29
1
Nayan wrote:

One question though,
It is time consuming to find whether 57 is the sum of any prime numbers.
I had to pick the primes below 57 and find out whether 2 primes can sum up to 57.
Is there any easy way?

57 is an odd number..
All prime numbers are odd except 2 ..therefore ODD+ODD=ODD is not possible
only possibility to check is 57=2+prime numb (which in this case is not possible)

This is the fastest way to do it.. Hope this Helps!
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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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01 Jan 2018, 21:55
x,y,z are +ve integers.. need to find whether x is a factor of z (an odd integer)

stmt1:
x,y => prime nos. whos sum is a factor of 57 (3 x 19) . Sum(x,y) cannot be 3 as it would mean one of x or y needs to be 1 (not prime)
Sum =19 & since x<y , the only possible pair of (x,y) = (2,17)

And since z is an odd integer x cannot divide z => Stmt 1 sufficient

Stmt2 : z is a factor of 57, but gives no information on x, y => not sufficient
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Re: If x, y, and z are positive integers such that x < y < z  [#permalink]

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16 Feb 2019, 08:39
mm007 wrote:
If x, y, and z are positive integers such that x < y < z, is x a factor of the odd integer z?

(1) x and y are prime numbers, whose sum is a factor of 57

(2) z is a factor of 57

x,y and z are +ive Integers.

x<y<z , is x a factor of integer z

from 1, if x and y are prime numbers , whose sum is a factor of 57

57 = 1*3*19

Now either we can get a 19, from sum of 2 prime integers

2<17<19, now 2 can never be a factor of odd number
Sufficient

from 2) z is a factor of 57, this can have 2 answers
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Re: If x, y, and z are positive integers such that x < y < z   [#permalink] 16 Feb 2019, 08:39
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