Question

What is the value of xyz?
Statement 1: x_a = y^b = z^c and ab + bc + ca 0 where a, b and c are non-zero integers
Statement 2: a^x = b, b^y = c, c^z = a where a, b and c are non-zero integers

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is D

The correct answer is option (D):

Either statement (1) alone or statement (2) alone is sufficient to answer the question

Let's analyze each statement to determine if it's sufficient to find the value of xyz.

Statement 1: x^a = y^b = z^c and ab + bc + ca = 0 where a, b and c are non-zero integers.

From x^a = y^b = z^c, we can't directly determine the relationship between x, y, and z. However, we are given ab + bc + ca = 0. While this equation holds, it doesn't give us a specific value for xyz. However, if x = y = z, and x^a = x^b = x^c, then a, b, and c would all be equal, implying a = b = c. The equation ab + bc + ca = 0 becomes a² + a² + a² = 0, or 3a² = 0. This means a = 0, which contradicts the information that a, b, and c are non-zero integers. Another possibility is x=y=z =1. Then 1^a = 1^b = 1^c =1. The value of xyz will be 1 * 1 * 1 = 1. If we can show that xyz must be 1, or that we know that they are related to each other such that their product is uniquely determined, then this will be sufficient. Note if x=y=z= -1, then (-1)^a = (-1)^b = (-1)^c = -1, and if a,b, and c are all odd numbers, then the equation ab+bc+ac=0 is not automatically satisfied. Therefore, Statement 1 alone is insufficient because it does not define xyz uniquely.

Statement 2: a^x = b, b^y = c, c^z = a where a, b and c are non-zero integers.

From a^x = b, b^y = c, and c^z = a, we can substitute. Let's start with a^x = b. Substitute b into b^y = c: (a^x)^y = c, which simplifies to a^xy = c. Now substitute c into c^z = a: (a^xy)^z = a, which simplifies to a^xyz = a¹. Assuming a ≠ 0 and a ≠ 1 and a ≠ -1, we can equate the exponents: xyz = 1. Therefore, Statement 2 alone is sufficient to find the value of xyz, which is 1.

Since statement 2 provides enough information to solve the question, we check if statement 1 can also be sufficient. We realize that statement 1 can't be solved as is, so the correct option must be that statement 2 is sufficient by itself. Note if the values are 1, then the products of xyz is 1 regardless of a, b, and c.

Therefore, the correct answer is: Either statement (1) alone or statement (2) alone is sufficient to answer the question.