## For nonnegative integers $$a, b,$$ and $$c,$$ what is the value of the product $$abc?$$

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### For nonnegative integers $$a, b,$$ and $$c,$$ what is the value of the product $$abc?$$

by Vincen » Thu Mar 24, 2022 12:46 pm

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## Global Stats

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

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

Source: Veritas Prep

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### Re: For nonnegative integers $$a, b,$$ and $$c,$$ what is the value of the product $$abc?$$

by [email protected] » Fri Mar 25, 2022 6:05 am

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## Global Stats

Vincen wrote:
Thu Mar 24, 2022 12:46 pm
For nonnegative integers $$a, b,$$ and $$c,$$ what is the value of the product $$abc?$$

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

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