Is {5^x}^{5-3x} = 1 ? 1) x is an Integer 2) x*y ≠ x

Question

Is ({5^x})^{5-3x} = 1 ? 1) x is an Integer 2) x*y ≠ x Source : www.GMATinsight.com

Archit3110 · Answer

target
Is  ({5^x})^{5-3x} = 1  ?
this is possible only when x=0 and x= 5/3
#1
x is an integer
we get yes and no ; insufficient
#2
x*y≠ x
only possible when x is any value except 0 and y is 0 or any other value we get no to target and yes to target when x=5/3 
insufficient
since x is an integer so x=0
sufficient 
option C

sk1999 · Answer

Hi, can somebody share a detailed solution for this one?

Frothygroundhog · Answer

First answer when does  (5^x)^{5-3x} =1 ?

(5^x)^{5-3x} = 5^{x(5-3x)}  -->  5^{x(5-3x)} = 5^{5x-3x^2}
 
The only way for  5^{5x-3x^2} = 1  is if the exponent  5x-3x^2 = 0

5x-3x^2 = 0  --> Add  3x^2  to both sides

5x = 3x^2  --> Divide both sides by  3x

x = 5/3  (this is 1 solution)

Another way to solve is to factor out x from  5x-3x^2 = 0

And you get  x(5-3x) = 0  and this is only possible is when  x = 0

Therefore  x = 0  or  x = \frac{5}{3}
 
 Statement (1)

Clearly insufficient as  x  could be any integer including  0

Statement (2)   x*y   ≠  x

First determine when  x*y = x  by re-writing the equation

x*y - x = 0  --> Now factor out x

x(y-1) = 0  --> this equation can only be true if  x = 0  or  y = 1

So we know from this statement that  x  cannot  = 0  as otherwise  x*y  will equal  0  which is equal to  x , but if  y = 1  then  x  could be any value including  \frac{5}{3}  as seen above. So this Statement alone is insufficient.

Combining Statement (1) and (2) 
From statement (1) we know that x  must be an integer  and could be any integer including  0 , and from statement (2) we know that integer cannot be  0 , and since we know the only way that  (5^x)^{5-3x} =1  is when  x = 0  or  x = \frac{5}{3} , then  (5^x)^{5-3x} ≠ 1 , thus giving us a definitive  NO  Answer, therefore combining both statements is sufficient to answer the question.

Answer: C.

Is {5^x}^{5-3x} = 1 ? 1) x is an Integer 2) x*y ≠ x

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Is {5^x}^{5-3x} = 1 ? 1) x is an Integer 2) x*y ≠ x

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards