If x is a positive integer greater than 2, the product of x consecutive positive integers must be divisible by which of

[BTGModeratorVI]

If x is a positive integer greater than 2, the product of x consecutive positive integers must be divisible by which of the following?

I. x - 1
II. 2x
III. x!

A. I only
B. II only
C. II and III only
D. I and III only
E. I, II and III only

Answer: E
Source: Economist GMAT

[Brent@GMATPrepNow]

If x is a positive integer greater than 2, the product of x consecutive positive integers must be divisible by which of the following?

I. x - 1
II. 2x
III. x!

A. I only
B. II only
C. II and III only
D. I and III only
E. I, II and III only

Answer: E
Source: Economist GMAT

----ASIDE------------------------
There's a nice rule says: The product of k consecutive integers is divisible by k, k-1, k-2,...,2, and 1
So, for example, the product of any 5 consecutive integers will be divisible by 5, 4, 3, 2 and 1
Likewise, the product of any 11 consecutive integers will be divisible by 11, 10, 9, . . . 3, 2 and 1
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.
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So, the product of x consecutive integers must be divisible by x, x-1, x-2, ....3, 2, and 1
So, we can immediately see that the product will be divisible by (x-1) and x! (since x! = (x)(x-1)(x-2).....(3)(2)(1)
So statements I and III are true

What about statement II?
We already know that the product of x consecutive integers must be divisible by x, x-1, x-2, ....3, 2, and 1
Since x > 2, then we know 2 and x are DIFFERENT numbers.
So, the product of x consecutive integers must be divisible x AND by 2.
This means the product must be divisible by 2x
So statement II is true

Answer: E

Cheers,
Brent

[Scott@TargetTestPrep]

If x is a positive integer greater than 2, the product of x consecutive positive integers must be divisible by which of the following?

I. x - 1
II. 2x
III. x!

A. I only
B. II only
C. II and III only
D. I and III only
E. I, II and III only

Answer: E
Source: Economist GMAT

Solution:

We can adhere to the rule that states that the product of x consecutive positive integers is divisible by x!. Thus, III is true. Additionally since 2 and x are distinct factors of x!, we see that x! is divisible by 2x; thus, II is true. Finally, since x - 1 will always be a factor of x!, then x! is divisible by x - 1. So, I is also true.

Answer: E