Problem Solving

If x is an integer, then x(x - 1)(x - k) must be evenly divisible by three when k is any of the following values EXCEPT:

(A) -4
(B) -2
(C) -1
(D) 2
(E) 5

Please explain your methodology.

EMAN wrote:If x is an integer, then x(x - 1)(x - k)ust be evenly divisible by three when k is any of the following values EXCEPT:

(A) -4
(B) -2
(C) -1
(D) 2
(E) 5

Please explain your methodology.

Just a note before we start: for some reason your formatting is messed up and the - sign is showing up as a bullet point instead.

So, just to restate the question:

...If x is an integer, then x(x - 1)(x - k)...

We know that x and (x-1) are consecutive integers.

We also know that any 3 consecutive integers contains a multiple of 3; therefore, the product of any 3 consecutive integers is divisible by 3.

So, in order for the whole product to be definitely divisible by 3, there are 2 possibilities:

1) k is just to the left of the first term or just to the right of the second term; and

2) k is a multiple of 3 distant from the possibilities mentioned in (1).

For (1), this basically means that k = -1 or +2, since:

(x-(-1)) = (x+1), which would give us (x+1), (x) and (x-1), which are consecutive; and

(x-(+2)) = (x-2), which would give us (x), (x-1) and (x-2), which are consecutive.

For (2), this means that:

k = -1 +/- (multiple of 3); or
k = +2 +/- (multiple of 3)

[which are actually the same condition].

So basically, if k = {..., -10, -7, -4, -1, 2, 5, 8, 11, ...} then we're good to go.

Taking a quick look at the choices, only -2 doesn't match our criteria, therefore (B) is the correct choice.

* * *

Now that we understand the concepts, let's think about how we could have answered this question merely by paying attention to our best friends, the answer choices.

The question asks about divisibility by 3; therefore, 3 is a very important number for this question.

Looking at the choices, A, C, D and E are separated by 3 each; only B deviates from this pattern.

If you're a fan of Sesame Street, you're probably familiar with the "One of these things is not like the others" game; on an EXCEPT question, if one answer is relevantly different from the others, it's almost certainly the correct choice.

Based on the pattern alone, and knowing that 3 is the king number in this question, we can confidently choose B without doing any math at all.