If x and y are positive integers, is x/y a terminating decimal?

Question

(1) y/x is even
(2) The greatest common factor of x and y is 1

M36-72

chetan2u · Accepted Answer

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

Bunuel · Answer

Official Solution:

If  x  and  y  are positive integers, is  \frac{x}{y}  a terminating decimal? 
 
We get a terminating decimal when a fraction,  reduced to its lowest term , has no primes other than 2 and/or 5 in the denominator. If a fraction,  reduced to its lowest term , has primes other than 2 and/or 5 in the denominator, the fraction will NOT terminate.
 
(1)  \frac{y}{x}  is even
 
  y=x*even=even .
 
  \frac{x}{y}=\frac{x}{x*even}=\frac{1}{even} . If that even number has no primes other than 2 and/or 5 in the denominator (for example, if that even number is 2, 4, 8, 10, 16, 20, 32, 40, ...), then the fraction will terminate but if that even number has primes other than 2 and/or 5 in the denominator (for example, if that even number is 6, 12, 14, 18, ...), then the fraction will NOT terminate. Not sufficient.
 
(2) The greatest common factor of  x  and  y  is 1
 
This one is clearly insufficient. Consider  x=1  and  y=2  for an YES answer and  x=1  and  y=3  for a NO answer. 
 
(1)+(2) From (1) we know that  x  is a factor of  y  and from (2) we know that only common factor they share is 1, so  x=1 . So, now we know that  x=1  and  y=even  but we still don't know whether  y  has primes other than 2 and/or 5. For example, if  x=1  and  y=2 , then the answer is YES but if  x=1  and  y=6 , then the answer is NO. Not sufficient.

Answer: E

Sateni7628 · Answer

Statement 1, y/x is even i.e 2 * (num), therefore x/y = 1/ (2*num). If this num is 5, x/y is terminating, if num is 7 x/y is not terminating. Not sufficient.

Statement 2, since GCF of x & y is 1, values of x & y could be 10 & 5 or 10 & 7. This statement is not sufficient since the x/y could be terminating or not terminating.

Together, the statements would not confirm whether the value of y is 5 or 7 or anything else.

Hence the answer is E. Correct me if I am wrong.

rahul16singh28 · Answer

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

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

GMATin · Answer

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

LoneSurvivor · Answer

x/y will be terminating if y is in the form of 2^a * 5^b
now s1: y= 2*p*x
Now whether x/y will be terminating depends upon P. So not sufficient. 
S2: Clearly not sufficient

S1+S2 x=1 but the rest depends upon P. So E

exc4libur · Answer

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  ,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)

If x and y are positive integers, is x/y a terminating decimal?

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If x and y are positive integers, is x/y a terminating decimal?

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