If ab + c = a(b + c), which of the following must be true?

Question

A. a = 1 and b = 0
B. a = 1 and c = 0
C. b = 1 and c = 0
D. a = 1 or c = 0
E. a = 1 or b = 0

I altered an official question to create this question in order to test whether the changes make the question harder or easier.

QMG · Answer

If we solve the equation, we get:

ab + c = a(b+c)
ab + c = ab + ac | -ab
c = ac

For this expression to be true, we either need c = 0 OR a = 1, so this leaves us with  choice D

BrentGMATPrepNow · Answer

Before I provide my solution, I think it's the good idea to review the difference between the SOLUTIONS to an equation, the VALUE of a variable in an equation.  
 For example, if (x - 3)(x + 4) = 0, then what conclusion can we draw about the VALUE of x? 
We can conclude that either  x = 3  OR  x = -4  (we use OR because x can't simultaneously have 2 different values)
On the other hand, the SOLUTIONS to the equation (x - 3)(x + 4) = 0 are  x = 3  AND  x = -4 , since each of those x-values satisfies the equation.

Now onto the question!!
Given:  ab + c = a(b + c) 
Expand the right side: ab + c = ab + ac
Subtract ab from both sides: c = ac
Subtract ac from both sides: c - ac = 0
Factor the left side:  c(1 - a) = 0

Key property: If AB = 0, then A = 0 or B = 0 (or they both equal 0)  
So, if  c(1 - a) = 0 , then  c = 0  or (1 - a) = 0, which means  a = 1

So, the correct answer is either B or D. 
We can eliminate answer choice B because it need not be the case that  c = 0  AND  a = 1 .
For example, c = 0 and a = 10 is a solution to the equation  c(1 - a) = 0 
Similarly, c = 3 and a = 1 is also a solution to the equation  c(1 - a) = 0

So, the correct answer is D.

minchakra · Answer

Dear expert, I am curious why the (E) can't not be the answer.
my understanding is when a=1, b=0
Then ab+c = a(b+c) >> (1)(0)+c = 1(0+C) >> c=c.

Thank you in advance

BrentGMATPrepNow · Answer

The key word here is  MUST .

While it's  possible  to have a solution in which a = 1 or b = 0, we can't say that it  must  be the case that either a = 1 or b = 0.
For example, the values a = 2, b = 2 and c = 0 satisfy the given equation. 
In this case, a ≠ 1 and b ≠ 0, which means answer choice E is not necessarily true.

Does that help?

BrushMyQuant · Answer

ab + c = a(b + c)

=> ab + c = ab + ac
=> ac - c = 0
=> c * (a - 1) = 0
=> Either c = 0 or a = 1

So,  Answer will be D 
Hope it helps!

If ab + c = a(b + c), which of the following must be true?

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If ab + c = a(b + c), which of the following must be true?

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