For nonnegative integers \(a, b,\) and \(c,\) what is the value of the product \(abc?\)

[Vincen]

For nonnegative integers \(a, b,\) and \(c,\) what is the value of the product \(abc?\)

(1) \(ab=bc\)
(2) \(a\ne c\)

Answer: C

Source: Veritas Prep

[Brent@GMATPrepNow]

For nonnegative integers \(a, b,\) and \(c,\) what is the value of the product \(abc?\)

(1) \(ab=bc\)
(2) \(a\ne c\)

Answer: C

Source: Veritas Prep

Target question: What is the value of the product abc?

Statement 1: ab = bc
Let's TEST some values.
There are several values of a, b and c that satisfy the condition that ab = bc. Here are two:
Case a: a = 0, b = 0 and c = 0. In this case, the answer to the target question is abc = (0)(0)(0) = 0
Case b: a = 1, b = 1 and c = 1. In this case, the answer to the target question is abc = (1)(1)(1) = 1
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: a≠c
Let's TEST some values.
Since we aren't told anything about the value of b, we cannot answer the target question with certainty.
So, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that ab = bc
Rewrite as: ab - bc = 0
Factor to get: b(a - c) = 0
This means that EITHER b = 0 OR (a - c) = 0

Statement 2 tells us that a≠c
So, it CANNOT be the case that a-c = 0
This means it MUST be the case that b = 0
If b = 0, then abc = 0
The answer to the target question is abc = 0
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent