For \(x/y\) to be a terminating decimal, y should be multiple of 2 or 5 only..

Let's see the statements..

1) y/x is even...
Just confirms that y is MULTIPLE of 2 and also of y..
But y can be 2, x/y will be terminating decimal
If y is 18 and X is 3, x/y will be 1/6, a non terminating decimal.
Insufficient

2) GCF of x and y is 1.
x as 3 and y as 2 will give x/y is terminating decimal.
x as 3 and y as 7 will give x/y as non-terminating decimal.
Insufficient

Combined
X and y are co-prime and y/x as even means x is 1 and y is MULTIPLE of 2..
If x is 1 and y is 10, x/y is terminating..
If x is 1 and y is 30, x/y is not terminating..
Insufficient

E
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We can do it by taking \(x = 1, y =6\) and \(x = 1, y = 4.\) It will stand for both the equations.
Hence, E

x/y will be a terminating decimal only if y is 1 or a multiple of 2 and/or 5. y cannot be a multiple of any other number other than 2 and/or 5.

With this principle in mind, let's look at the statements

Statement 1: y/x is even. So, this means either x=1 and y=even number or x is a factor of y such that the resulting fraction y/x is even
Let's take examples to understand this.

If x=3 and y=6, then y/x=6/3=2(=even number). But since y=6, x/y can't be terminating decimal

If x=5 and y=10, then y/x=10/5=2(=even number). Since y=10, x/y is a terminating decimal

Thus, this statement is insufficient

Statement 2: If the greatest common factor between two numbers is 1, this proves that any of the following cases can be true: 1) the two numbers are odd and even or 2) the two numbers are prime number or 3) that at least one of the two numbers is 1.

In any of the above cases, we cannot distinguish which one will y be, hence we cannot be sure if x/y will be terminating or not.

Thus, this statement is insufficient

Combining statements 1 & 2: we get that contrary to what we had deduced in statement 1, x cannot be a factor of y. Thus, x=1 and y is an even number.

Checking on the three cases:

1) x=1 and y= even doesn't prove whether x/y will necessarily be terminating. Eg: x=1 and y=10: results in x/y terminating vs x=1 and y=14, results in x/y non-terminating
Although this is enough to prove that statement 2 is insufficient, let's check the other two cases as well for our understanding.

2) This cannot hold true since x=1 which is neither prime nor composite

3) This is a repeat of case 1.

Thus, the statements when combined are also insufficient

Answer is E

Terminating decimal: one in which the denominator is made of only 2s, 5s or both (ie. 1/2 1/5 1/10);

Divisibility O & E:
E/E = (E,O,Fraction)
O/O = (O,Fraction)
E/O = (E,Fraction)
O/E = (Div by 0 [NA],Fraction)

(1) y/x is even: insufic.

y/x=even… E/E or E/O;
y/x=12/3=4; y/x=3/12=1/4; answer yes.
y/x=12/2=6; y/x=2/12=1/6; answer no.
y/x=12/1=12; y/x=1/12; answer no.

(2) The greatest common factor of x and y is 1: insufic.

x,y=co-prime
x/y=1/2; answer yes.
x/y=3/7; answer no.

(1&2): insufic.

x,y=co-prime, and y/x=EVEN then y/x≠E/E=E/O;
so if y=EVEN then x=ODD=1
y/x=10/1=10; y/x=1/10; answer yes
y/x=12/1=12; y/x=1/12; answer no

Answer (E)