Number Properties

[singhmaharaj]

If xy + z = x(y+z), which of the following must be true?

1) x=0 and z = 0

2) x=1 and y = 1

3) y=1 and z = 0

4) x=1 or y=0

5) x=1 or z=0

[Brent@GMATPrepNow]

If xy + z = x(y+z), which of the following MUST be true?

1) x=0 and z = 0

2) x=1 and y = 1

3) y=1 and z = 0

4) x=1 or y=0

5) x=1 or z=0

The key word here is must.

Given: xy+z = x(y+z)
Expand: xy+z = xy + xz
Subtract xy from both sides: z = xz
Rearrange: xz - z = 0
Factor: z(x-1) = 0

This tells us that z = 0 or x = 1
Answer: E

Cheers,
Brent

[ceilidh.erickson]

Alternatively, instead of doing algebra we could just ask for each answer choice: is there a case when this would NOT be true?

Given that xy + z = x(y+z), try to prove each answer choice WRONG:

1) x=0 and z = 0
Does x have to be 0? No, x could be 2, y could be 3, and z could be 0: 2*3 + 0 = 2(3 + 0). Wrong.

2) x=1 and y = 1
Same as above: x could be 2. Wrong.

3) y=1 and z = 0
Using the same example from earlier, y doesn't have to be 1. Wrong.

4) x=1 or y=0
Using the same example, x doesn't have to be 1 or y = 0. Wrong.

5) x=1 or z=0
Can we think of an example in which x is not 1 and z is not 0? No, there's no possibility that works. Correct!