If x and y are positive integers is y odd?(1) (y+2)!/x! = odd. Notice that
\frac{(y+2)!}{x!}=odd can happen only in two cases:
A.
(y+2)!=x! in this case
\frac{(y+2)!}{x!}=1=odd. For this case
y can be even:
y=2=even and
x=4:
\frac{(y+2)!}{x!}=\frac{24}{24}=1=odd;
B.
y=odd and
x=y+1, in this case
\frac{(y+2)!}{(y+1)!}=y+2=odd. For example,
y=1=odd and
x=y+1=2:
\frac{(y+2)!}{x!}=\frac{3!}{2!}=3=odd (basically all the cases like: 3!/2!, 5!/4!, 7!/6!, ... --> y+2=odd, 3, 5, 7, ... --> y=odd, 1, 3, 5, ...).
Not sufficient.
(2) (y+2)!/x! is greater than 2. Clearly insufficient: consider
y=1=odd and
x=2 OR
y=2=even and
x=1.
(1)+(2) From (2) we cannot have case A, hence we have case B, which means
y=odd. Sufficient.
Answer: C.
Hope it's clear.
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