If a, b, c and d are integers, is (2^a·3^b)/(2^c·3^d) even ?

Question

If a, b, c and d are integers, is  \frac{2^a*3^b}{2^c*3^d}  even?

(1) a > c
(2) b = c + d

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$!vakumar.m · Answer

If a, b, c and d are integers, is  \frac{ }{ }  even?
(1) a > c
no info on b and d is given
not sufficient
(2) b = c + d
the ratio can be re-written as
  \frac{ }{ }

=>  \frac{ }{ }

=> Clearly, the ratio is an Even integer only if a >c (and b>c) about which we don't have any info from statement 2 alone.
Statement 2 is not sufficient.

statement 1 and 2 taken together - a>c and b = c+d. 
if a = 2 and c = 1 - then the ratio is even.
if a =2, and c = -1 - then the ratio is a fraction. 
Not sufficient. 
Answer E

Nabneet · Answer

For Condition a>c 
if a=2 and c=-2

sivakumarm786

$!vakumar.m · Answer

Thanks for pointing out the mistake. I had read the question as a, b, c and d to be positive integers which is not so.
Since, a, b, c and d are integers we need to know their signs as well. So, E shall be the answer.

bv8562 · Answer

If a, b, c and d are integers, is  \frac{2a∗3b}{2c∗3d }even?

(1) a > c
(2) b = c + d

2^a-c * 3^b-d

1) a > c => a-c > 0

This means we have 2 with a positive exponent. But we still do not know the whether the exponent of 3 is positive or negative. If its positive then we will have an even integer. But if its negative then we will have a fraction.

2) b = c + d

2^a-c * 3^b-d = 2^a-c * 3^c+d-d = 2^a-c * 3^c

In this case we do not know the sign of exponent of neither 2 nor 3. If a-c is positive and C is positive then we will have an even integer. But if a-c is positive and c is negative then we will have a fraction.

3) a > c & b = c + d

By combining both statements we are left with following expression: 2^a-c * 3^c

We still do not know the sign of c. If c is positive then we will have an even integer. But if c is negative then we will have a fraction.

Therefore, its E.

bumpbot · Answer

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If a, b, c and d are integers, is (2^a·3^b)/(2^c·3^d) even ?

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