BTGModeratorVI wrote: ↑

Sun Jul 26, 2020 6:44 am

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.
----------------------------------

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