If a+b+c = 0 and a^3+b^3+c^3 = 216, What is the Value of a∗b∗c ?

Question: If a+b+c = 0 and a^3+b^3+c^3 = 216, what is the value of a∗b∗c ?

1. -72
2. 48
3. 72
4. 24
5. 216

“If a+b+c = 0 and a^3+b^3+c^3 = 216, what is the value of a∗b∗c?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of 31 questions in total. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution 1:

It is given in the question that a+b+c = 0 and a^3+b^3+c^3 = 216.
It has asked to find out the value of a * b * c.
Let the first equation is a+b+c = 0
And the second equation is a^3+b^3+c^3 = 216
From (1) we get,
a+b+c = 0
a+b = -c
Raising both sides to third power,
\((a+b)^3 = (-c)^3\)
\(a^3 + 3a^2b + 3ab^2 + b^3 = -c3\)
\(a^3 + b^3 + c^3 + 3a^2b + 3ab^2 = 0\)
\(a^3 + b^3 + c^3 + 3ab(a+b)= 0\)
\(a^3 + b^3 + c^3 + 3ab(-c)= 0\) (putting the value of a+b)
216 - 3 (a*b*c) = 0
a*b*c = 216/3 = 72

Therefore C is the correct answer.

Approach Solution 2:

It is given in the question that a+b+c = 0 and a^3+b^3+c^3 = 216.
An alternate way would be,
Since we are given answer options a*b*c
We can find the factors of given answer options also given is that a+b+c=0 which means that either the integer values are two negative and one positive or one negative and two positive.
lets start from 72
Factors of 72

\(72 = 2^3 + 3^2\)

which can be ; 9 - 8 - 1=0
and its cube ; 9^3 - 8^3 - 1 = 216
This satisfies our condition

so a*b*c = 72

Approach Solution 3:

We know that:
a³ + b³ + c³ - 3abc = (a+b+c)(a²+b²+c²-ab-bc-ac)

However, a+b+c=0

=> a³+b³+c³-3abc = 0
=> a³+b³+c³= 3abc
=> 3abc = 216
=> abc = 72

C is correct option

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