If x–1 is a positive integer, is x–1 divisible by 3?

Question

(1) (x^2 + x)(x + 1) is divisible by 3.

(2) x(x^2–x) is divisible by 3.

Ljchen121 · Accepted Answer

Thanks Vaibhav. This was my train of thought: just factor everything
1) x(x+1)(x+1) , either (x+1) or x must be divisible by 3, so (x-1) is not divisible by 3. SUF
2) (x)(x)(x-1), x or (x-1) must be divisible by 3, but which one, it is uncertain INS

narangvaibhav · Answer

Hi Ljchen

use substitution for both the statements..

best Vaibhav

anordinaryguy · Answer

Ljchen121,

this is about three consecutive positive (x-1 > 0 => x > 1) number x-1, x, x+1. It is evident, only one of these can be divisible by 3

So from Statements : 
(1) (x^2 + x)(x + 1) is divisible by 3.
=> x (x+1) (x+1) is divisible by 3.
=> Either x OR x+1 is divisible by 3.
=> It implies (x-1) can't be divisible by 3
=> SUFFICIENT

(2) x(x^2–x) is divisible by 3. 
=> x (x) (x-1) is divisible by 3.
=> Either x OR x-1 is divisible by 3.
=> It implies (x-1) may or may not be divisible by 3
=> NOT-SUFFICIENT

Answer is A

Bunuel · Answer

That's a correct approach.

If x–1 is a positive integer, is x–1 divisible by 3?

Notice that x–1 is an integer means that x is an integer.

(1) (x^2 + x)(x + 1) is divisible by 3 --> x(x+1)^2 is divisible by 3 --> either x or x+1 is divisible by 3. Now, out of three consecutive integers x-1, x, and x+1 ONLY ONE is divisible by 3 and since we know that it's either x or x+1 then x-1 is definitely not divisible by 3. Sufficient.

(2) x(x^2–x) is divisible by 3 --> x^2(x-1) is divisible by 3 --> either x or x-1 is divisible by 3, but we don't know which one. Not sufficient.

Answer: A.

Helium · Answer

This is a Yes/No question, so either a definite “yes” or a definite “no” would be Sufficient. We are told that x – 1 is an integer greater than 0, so x must be an integer greater than 1. We need to know whether the expression x – 1 is definitely divisible by 3, which means that when x — 1 is divided by 3, the result is an integer. This will be so if and only if x – 1 is a multiple of 3.

Evaluate the Statements:

Statement (1): While we could FOIL ( x2  + x)(x + 1), doing so would produce something even more complicated. It’s simpler to factor out the x common to both terms in (x2 + x):

( x2  + x)(x + 1)

x(x + 1)(x + 1)

Statement (1) tells us that x(x + 1)(x + 1) is divisible by 3, which means that either x or x + 1 has to be a multiple of 3.

Multiples of 3 only come along every third integer. So if x is a multiple of 3, then x – 1 definitely can’t be (x– 3 would be, but not x – 1). Similarly, if x + 1 is a multiple of 3, then again x – 1 definitely can’t be (x – 2 would be, but not x – 1). Since x – 1 is definitely not a multiple of 3, this Statement is Sufficient.

Picking Numbers can help us to illustrate this. If we let x = 3, then x – 1 = 2, which answers our question with a “no”; 2 is not divisible by 3. If we pick x + 1 = 3, then x – 1 = 1. This also gives us a “no,” as 1 is not divisible by 3.

With always producing a “no,” this Statement is Sufficient to answer the question, and we can eliminate choices (B), (C), and (E).

Statement (2): As before, we can simplify the given expression by factoring:

x( x2 –x)

x(x)(x– 1)

Using the same reasoning as before, we know that if this expression is divisible by 3, then either x or x – 1 is divisible by 3. If x is divisible by 3, then x – 1 will not be. In other words, the expression x – 1 may or may not be divisible by 3. This statement is therefore Insufficient to answer the question. Eliminate choice (D).

Therefore, Answer Choice (A) is correct.

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If x–1 is a positive integer, is x–1 divisible by 3?

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If x–1 is a positive integer, is x–1 divisible by 3?

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