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

Bunuel- you stated that "Either b =0 or a = c." when we get to the point of b(a-c)=0 is that always the case, where only one is true, when we have an equation that looks like that? So for example (just making this up)...

12A=9AB
12A-9AB=0
factor out the 3A...
3A(4-3B)=0
So then..only one of these solutions can be true?
A=0 OR B=4/3

Hi soniasawhney,

Both possibilities COULD be true, but it's not grammatically correct (nor mathematically correct) to state:

B=0 AND A=C (as both DO NOT need to be true for the product of B(A-C) to equal 0 - only one of them needs to be true).

If you were taking the notes, then you might find it useful to write....

B=0 or A=C OR BOTH

In your example:
12A = 9AB
12A - 9AB = 0
3A(4 - 3B) = 0

A=0 or B=4/3 OR BOTH

GMAT assassins aren't born, they're made,
Rich

ah that makes sense- thank you!

(1) ab=bc

\(ab = bc\)

\(ab - bc = 0\)

\(b(a - c) = 0\)

Either \(b = 0\) or \(a = c\)

We will get multiple answers based on values of b and a or c.

If b = 0 then we will the value of equation as ZERO, however if the a = c then we will get value, based on a or c.

Hence, (1) ===== is NOT SUFFICIENT

(2) a≠c

This does not give us any values of a or b or c

Hence, (2) ===== is NOT SUFFICIENT

Combining (1) & (2)

We get that a≠c which means \(b = 0\) as per (1) and this gives us the value of ZERO foe abc.

Hence, (1) & (2) ===== is SUFFICIENT

Hence, Answer is C

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There are 3 variables and 0 equation. Thus the answer E is most likely.

Condition 1)
\(ab = bc\) is equivalent to \(ab - bc = 0\) or \(b(a-c) = 0\).
Thus \(b = 0\) or \(a = c\).
If \(b = 0\), then we have \(abc = 0\).
However, if \(b \ne 0\) and \(a = c\), we have the case \(a = b = c = 1\) with \(abc = 1\).
Since the answer is not unique, this is not sufficient.

Condition 2)
If \(a = 1\), \(b = 1\), \(c = 2\), then \(abc = 2\).
If \(a = 1\), \(b = 2\), \(c = 3\), then \(abc = 6\).

Since the answer is not unique, this is not sufficient either.

Conditions 1) & 2)
From the condition 1), \(ab = bc\) is equivalent to \(ab - bc = 0\) or \(b(a-c) = 0\).
Thus \(b = 0\) or \(a = c\).
However, \(b = 0\), since \(a \ne c\) by the condition 2)
Thus \(abc = 0\).

Therefore, C is the answer.


For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80 % chance that E is the answer, while C has 15% chance and A, B or D has 5% chance. Since E is most likely to be the answer using 1) and 2) together according to DS definition. Obviously there may be cases where the answer is A, B, C or D.

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

What is the value of product abc?

(1) \(ab = bc\)

If b = 0, then a and c could be any value. INSUFFICIENT.

(2) Clearly insufficient.

(1&2) If \(ab = bc \)and \(a≠c\), b must be zero. Therefore abc = 0. SUFFICIENT.

Answer is C.

Solution:

Statement One Alone:

ab = bc

If a = 1, b = 0, and c = 2, then abc = 0. However, if a = 2, b = 1, and c = 2, then abc = 4. Notice that ab = bc is satisfied in both cases. Since we could have more than one possible value for abc, statement one alone is not sufficient.

Statement Two Alone:

a ≠ c

Since we don’t know the value of b, we can’t determine the value of abc. Statement two alone is not sufficient.

Statements One and Two Together:

Let’s rewrite the equality as ab - bc = 0. Factoring out b, we have b(a - c) = 0. Using the zero product property, this can only be true if b = 0 or a - c = 0. Since a ≠ c, a - c ≠ 0. In other words, b must be 0. If b is 0, then regardless of the values of a and c, we will have abc = 0.

Answer: C

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