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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gmatophobia · Answer

\frac{2^a*3^b}{2^c*3^d}

2^{a-c}*3^{b-d}

For the fraction to be an integer

a \geq{c}  
  b \geq{d}

For the expression to be even

a > c  
  b \geq{d}

Statement 1

(1) a > c

We don't have information if  b \geq{d}

Hence the statement is not sufficient

Statement 2

(2) b = c + d

We don't have information if  a > c

Hence the statement is not sufficient

Combined

As we do not know the positive or negative nature of c (or the other variables), we can't comment on whether b > d

Hence the statements combined are not sufficient.

Option E

Suvankar8250 · Answer

The Fraction needs to be an integer first and then we can think about it being Positive even integer.
To make the Fraction integer there are 2 necessary conditions, i) a >= c , ii) b >= d
To make the Fraction an even integer the necessary condition is, i) a > c , since anything multiplied by an even integer (2) is even.

Statement 1 : Even though the statement ensures that the fraction might be even, it doesn't assure that the fraction will be an integer since no information about b >= d has been provided. Therefore, the statement is insufficient.

Statement 2 : b = c + d doesn't necessarily ensure that b >= d. Please note that a,b,c,d are integers and not positive integers. Lets say if 'c' and 'd' are negative integers, then b < d, for example, b = (-3) + (-2) = -5. This statement also doesn't provide any information about whether a >= c or not. Therefore, the statement is insufficient.

Together : Again b = c + d doesn't necessarily ensure that b >= d; an example has been given above. Even though now we know a > c , but the relation between 'b' and 'd' is still unknown to us, and all the scenarios are possible : b>d, b=d, b<d.

Therefore, both these statements together are insufficient and hence the answer is Option E.

If a, b, c and d are integers, is (2^a3^b)/(2^c3^d) even?

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