exponents

D. Either statement by itself is sufficient.

(4^x)^(5-3x)=1 can be rewritten...
(4^x(5-3x))=1
If 4 to a power is equal to 1, the power must equal 0 (as any value to the 0 power is equal to 1).
So, x(5-3x) = 0.
Two solutions:
x = 0 or x = 5/3

(1) x is an integer.
Sufficient. We have two possible solutions and only one is an integer (0).

(2) The product of x and positive integer y is not x.
Sufficient. Plug in the values. 0 x any positive integer is always 0. We need a non-zero value of x for this to be true. And, we have one (5/3).

@BuckeyeT

Thanks for the explaination. But, for GMAT DS questions statements will/should always yield same value for the given variable. In this case, if we use statement 1. we get X=0 and if we use statement 2 we get x= 5/3 which can't be correct.

I think the question is not properly structured. Can someone advise us on this?

Regards,
hhk

Hariharakarthi-

I agree. I haven't come across any like this. But, I couldn't find any issue with my algebra. And under the letter of the law, either statement would be correct by itself.

I'll be interested to hear if there is a correction for my work or for the question itself.

Thanks!

@BuckeyeT

I am clear and agreed with your explanation. My only doubt is on validity of the question?

Regards,
hhk

BuckeyeT wrote:D. Either statement by itself is sufficient.

(4^x)^(5-3x)=1 can be rewritten...
(4^x(5-3x))=1
If 4 to a power is equal to 1, the power must equal 0 (as any value to the 0 power is equal to 1).
So, x(5-3x) = 0.
Two solutions:
x = 0 or x = 5/3

A very common mistake that people make in data sufficiency, especially on complicated algebraic questions, is forgetting the difference between the question and factual information.

All of your math is good up to this point, but you forgot that you're dealing with a question. We don't have two solutions for x; we have two different values of x that would generate a "yes" answer to the original question.

A great thing to do to avoid making this mistake is, when you do your scratchwork, keep writing the word "is" at the beginning of each manipulation and putting a question mark at the end - that way you won't forget that you're dealing with a question.

So, after all of your work, we can restate the question:

Does x = 0 or 5/3?

To the statements:

(1) x is an integer.

well, we're allowed to pick x=0, because that fits the statement. As shown by the work, if x=0, we get a "yes" answer.

However, we're also allowed to pick any other integer, all of which return "no" answers to the original question.

We can get a yes and a no: insufficient.

(2) Fancy way of saying that x isn't 0.

If x does not equal 0, we can still pick x = 5/3 to get a "yes" answer and any other non-0 number to get a "no" answer. Since we can get both a yes and a no, insufficient.

Combined:

From (1), we know that x is an integer; from (2) we know that x isn't 0.

So, we know that x is a non-0 integer.

Therefore, x can be neither 0 nor 5/3 and we get a definite "no" answer to the original question.

Together sufficient, apart insufficient: choose C.

Stuart,

How can we say that does x = 0 or 5/3?

The question does not says that (4^x)^(5-3x)=1. The question is whether (4^x)^(5-3x)=1?

Please clarify.

Stuart Kovinsky wrote:

BuckeyeT wrote:D. Either statement by itself is sufficient.

(4^x)^(5-3x)=1 can be rewritten...
(4^x(5-3x))=1
If 4 to a power is equal to 1, the power must equal 0 (as any value to the 0 power is equal to 1).
So, x(5-3x) = 0.
Two solutions:
x = 0 or x = 5/3

A very common mistake that people make in data sufficiency, especially on complicated algebraic questions, is forgetting the difference between the question and factual information.

All of your math is good up to this point, but you forgot that you're dealing with a question. We don't have two solutions for x; we have two different values of x that would generate a "yes" answer to the original question.

A great thing to do to avoid making this mistake is, when you do your scratchwork, keep writing the word "is" at the beginning of each manipulation and putting a question mark at the end - that way you won't forget that you're dealing with a question.

So, after all of your work, we can restate the question:

Does x = 0 or 5/3?

To the statements:

(1) x is an integer.

well, we're allowed to pick x=0, because that fits the statement. As shown by the work, if x=0, we get a "yes" answer.

However, we're also allowed to pick any other integer, all of which return "no" answers to the original question.

We can get a yes and a no: insufficient.

(2) Fancy way of saying that x isn't 0.

If x does not equal 0, we can still pick x = 5/3 to get a "yes" answer and any other non-0 number to get a "no" answer. Since we can get both a yes and a no, insufficient.

Combined:

From (1), we know that x is an integer; from (2) we know that x isn't 0.

So, we know that x is a non-0 integer.

Therefore, x can be neither 0 nor 5/3 and we get a definite "no" answer to the original question.

Together sufficient, apart insufficient: choose C.

mehravikas wrote:Stuart,

How can we say that does x = 0 or 5/3?

The question does not says that (4^x)^(5-3x)=1. The question is whether (4^x)^(5-3x)=1?

Please clarify.

Here's the question:

Does (4^x)^(5-3x)=1?

We can simplify the question by simplifying the math (I'm stealing the math directly from BuckeyeT), but we always keep it in the form of a question:

Does (4^x(5-3x)) = 1 ?

We know that if 4 to a power is equal to 1, the power must equal 0 (as any value to the 0 power is equal to 1).

So, we can now rephrase the question to:

Does the exponent = 0?

Does x(5-3x) = 0?

When will that be true? When x = 0 or x = 5/3. So, our final question is:

Does x = 0 or 5/3?

If we get a yes answer to the final question, we get a yes answer to the original question; if we get a no answer to the final question, we get a no answer to the original question.

Thanks Stuart