mparakala wrote:Source: Beat the GMAT (this question was not answered by experts and hence, the repost)
If x > 0, then is x^3 - 3(x^2) + 2x divisible by 4?
1. x = 4y + 4, where y is an integer
2. x = 2z + 2, where z is an integer
I think the answer is A. I do no know the correct answer. Can any expert explain in detail?
Thank you!
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Target question:
Is x^3- 3x^2 + 2x divisible by 4?
Notice that the expression can be factored as (x-2)(x-1)(x) [aside: I wrote the terms in this order for a reason, as you'll see]
Rephrased Target question:
Is (x-2)(x-1)(x) divisible by 4?
IMPORTANT: notice that x-2, x-1 and x are three consecutive integers.
Question: Under what conditions will the product of 3 consecutive integers be divisible by 4?
This will occur if one of the 3 integers is divisible by 4
or if the first and last numbers (x-2 and x) are even.
If the x and x-2 are even (i.e., divisible by 2) then (x-2)(x-1)(x) must be divisible by 4
Onto the statements.
Given: x = 4y + 4, where y is an integer
Factor to get: x = 4(y + 1)
This tells us that x is divisible by 4, which means x is even
If x is even, then (according to my point in
green above)
(x-2)(x-1)(x) must be divisible by 4
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: x = 2z + 2, where z is an integer
Factor to get: x = 2(z + 1)
This tells us that x is divisible by 2, which means x is even
If x is even, then (according to my point in
green above)
(x-2)(x-1)(x) must be divisible by 4
Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer =
D
Cheers,
Brent
