If x, y, and z are positive integers such that x < y < z is x a factor

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?

My answer is C

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

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.

amit.

dr shpak - it doesn't say they are consecutive numbers.

My initial reaction to this was E and I agree with Hobbit.

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.

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.


Nothing about z ............ INSUFF

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

(C) is my answer. Please correct me if I'm wrong.

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.

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

Answer is E

Where am I going wrong here? How is the answer A?

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.

Please press Kudos if you find the question helpful!

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!!

Correct answer would be A

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!

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

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

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

Question: is x a factor of the odd integer z?

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

Factors of 57 = {1, 3, 19, 57}
x+y = sum of two prime Numbers ≠ 1
x+y = sum of two prime Numbers ≠ 3
x+y = sum of two prime Numbers = 19 if x=2 and y = 17
x+y = sum of two prime Numbers ≠ 57

i.e. x will be 2 and x can NOT be a factor of ODD integer z hence

SUFFICIENT

Statement 2: z is a factor of 57

No information about x hence

NOT SUFFICIENT

Answer: Option A

YOU CAN SOLVE THIS QUESTION WITHOUT PEN AND PAPER.
THIS IS HOW
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
The factor of 57 is 1,3,19, and 57.
1 & 3 can't be the sum of x&y as they can't be the sum of two prime no.
one imp thing before checking for 19&57 is "the sum of any two prime numbers is even unless it includes 2 in it(every prime no is odd and odd+odd=even, '2' is the only prime no which is even)
so 2 must be one of the prime numbers.
check for 19....2+17 can be 19 and both are prime numbers.
for 57.... 2+55 can be 57 but 55 is not a prime number, so reject it.
x=2, y=17, and since Z is an odd no as given in the question.
X can't be the factor of Z
Sufficient



(2) z is a factor of 57

information about x & y is not given,
so Insufficient.

Hence answer is A


If you still have any queries to solve a question without writing it on paper.
Send it to me I will try to solve it as a short method as possible.

Visualise all 3 on a number line from left to right. Important to note: z is odd.
Is x a factor of z?

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

57 is odd. It's factor must be odd only. If sum of two prime numbers is odd, one of those two prime numbers MUST BE 2, the only even prime number and the smallest one. Since x < y, x is 2.
2 cannot be a factor of an odd number z. Hence answer is definitely NO.
Sufficient Alone.

(2) z is a factor of 57

A factor of 57, z could be 57 (simplest case) so x could be 1 or 2 or 9 or 10 etc.
If x is 1, x will be a factor of z but if x is 10, x will not be a factor of z.
Not sufficient.

Answer (A)

Check out this video on Factors: https://youtu.be/DxIH8rjhpKY
and these posts:
https://anaprep.com/number-properties-f ... -a-number/
https://anaprep.com/number-properties-r ... e-factors/