If x, y, and z are positive integers, where x y z, is x*y*z an odd

Question

If x, y, and z are positive integers, where x > y > z, is x*y*z an odd number ?

(1) p - z is an odd number, where p is a prime number
(2) x + 1 is a prime number

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yashikaaggarwal · Answer

(1) p - z is an odd number, where p is a prime number
Case 1: P = 2
Z = 1
No value for X and Y to determine X*Y*Z is odd or even.

Case 2: P = 3
Z = 2
Irrespective of any value for X and Y, X*Y*Z will be even.
(Insufficient)

(2) x + 1 is a prime number
X, Y, and Z are positive no., and X>Y>Z
so the minimum value of X will be 3
but 3+1 = 4 (not a prime no. and in order for X+1 to be a prime number)
X has to be even
so X*Y*Z will be even.
(Sufficient)

Answer is B

mdsaddamforgmat · Answer

IMO B
For a product of three numbers to be odd, all the numbers should be odd

Statement1
From equation 1 we can say that Z is odd but cannot say anything about x & Y
Hence, insufficient

Statement2
Here, we can say X is even as even+odd = odd
If X is even then no matter whether z & Y are odd or even, overall multiplication will always give even result.
Hence, sufficient to answer.

frroldao · Answer

IMO, B

1)  Not suf 
- Case p = 2, z needs to be odd (e.g., 1)
- Case p=3, z needs to be even (e.g., 2)

2) if  x+1=prime , the  only prime number that by adding another prime results in another prime is 2 . So x = 2. Having this, whatever numbers are y and z,  x*y*z will always be even due to having x= 2. Hence, Sufficient

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kntombat · Answer

yashikaaggarwal  I have one question.
Statement 2 states that x+1 is a prime number, based on your approach you concluded that x has to be even which I agree with but the question I have is the smallest prime number is the digit 2, so if the value of x is 1 then 1+1 is also a prime number then shouldn't statement B also be insufficient ?

yashikaaggarwal · Answer

See, X can't be 1 and 2
why?
Simply because X, Y, and Z are positive integers.
the smallest positive integer is 1
and Z is the smallest among X, Y, and Z.
so taking the least value of X, Y, and Z
Z is 1, Y is 2, and X is 3
Now we know, X+1 is a prime.
and the least value of X = 3+1 = 4 is not a prime.
so we know, every prime no. - 1 is even no. (except 1, which can't be the value of X because of given constraint)
so X will be an even no.
and any multiple of even is even.
so X*Y*Z is even.
hope its clear.

kntombat · Answer

Yes now it's clear. Thank you  yashikaaggarwal .

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If x, y, and z are positive integers, where x > y > z, is xyz an odd

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If x, y, and z are positive integers, where x > y > z, is x*y*z an odd