intergers:

[\'manpreet singh]

If x is a positive integer, is x3- 3x2 + 2x divisible by 4?

(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

[anuprajan5]

The answer is D

x3- 3x2 + 2x breaks up into x(x-1)(x-2)

Statement 1 says that x is divisible by 4 - Sufficient. Then x(x-1)(x-2) is divisible by 4.
Statement 2 says that x is an even number. In a consecutive set of 3 integers, if x is even, then x-2 is also even, thereby implying 2 2s. Therefore sufficient.

[Brent@GMATPrepNow]

If x is a positive integer, is x^3- 3x^2 + 2x divisible by 4?

(1) x = 4y + 4, where y is an integer
(2) x = 2z + 2, where z is an integer

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

[anuprajan5]

If x is a positive integer, is x^3- 3x^2 + 2x 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

Hey Brent,

A quick one. Since the above bold says that x is divisible by 4, doesnt that already mean that a product of x and any other numbers will also be divisible by 4. It's just overcomplicating things a wee bit when we have to look at the series and then conclude that x-2 is also even.

[Brent@GMATPrepNow]

If x is a positive integer, is x^3- 3x^2 + 2x 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

Hey Brent,

A quick one. Since the above bold says that x is divisible by 4, doesnt that already mean that a product of x and any other numbers will also be divisible by 4. It's just overcomplicating things a wee bit when we have to look at the series and then conclude that x-2 is also even.

Ah, you're totally right! I kind of took a step backwards there, didn't I?
In fact, earlier in the solution, I said that the product of 3 consecutive integers be divisible by 4 if one of the 3 integers is divisible by 4. Then, after I learn that one of the integers is divisible by 4, I take the longer/slower approach to reach the same conclusion.

Good catch!

Cheers,
Brent

[\'manpreet singh]

Yes Anup you are right! Just by concluding that x is divisible by 4 is enough for us, but thanks anyways Brent for going deep into the concept , its was good brain exercise for me haha.


Singh.