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Re: If x, y, and z are positive integers such that x < y < z [#permalink]
27 May 2012, 03:57

6

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Expert's post

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.

Re: DS_If x, y, and z are positive integers... [#permalink]
27 May 2012, 00:55

2

This post received KUDOS

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.

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?

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

this leads us to nothing as I know nothing about z
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

Thus, the answer is C

Is it helpful, the way I expain ? or is too much details and too confusing ?

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.

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.

Re: If x, y, and z are positive integers such that x < y < z [#permalink]
09 Feb 2014, 04:53

A is the answer.

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
[#permalink]
09 Feb 2014, 04:53