If x is the product of three consecutive positive integers,

BTGModeratorVI wrote: ↑

Sat Feb 15, 2020 3:28 pm

If x is the product of three consecutive positive integers, which of the following must be true?

I. x is an integer multiple of 3.
II. x is an integer multiple of 4.
III. x is an integer multiple of 6.

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

Source: Official Guide
Answer: D

Let's take each statement one by one.

I. x is an integer multiple of 3: Among any three consecutive integers, there is at least one integer that is a multiple of 3. True.

II. x is an integer multiple of 4: Not must be true. Example: 1, 2 and 3.

III. x is an integer multiple of 6: Among any three consecutive integers, there is at least one integer that is a multiple of 2 and there is at least one integer that is a multiple of 3. So, x is an integer multiple of 2*3 = 6. True.

The correct answer: D

Hope this helps!

-Jay
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BTGModeratorVI wrote: ↑

Sat Feb 15, 2020 3:28 pm

If x is the product of three consecutive positive integers, which of the following must be true?

I. x is an integer multiple of 3.
II. x is an integer multiple of 4.
III. x is an integer multiple of 6.

(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II, and III

Source: Official Guide
Answer: D

----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 the divisors (noted above) are guaranteed.
------------------------------

Since x = the product of three consecutive integers, the above rule tells us that x is definitely divisible by 3 and 2.
So statement I is true

Also, if x is divisible by 3 and 2, then x is also divisible by 6.
So statement III is true

What about statement II? Is x necessarily divisible by 4?
No. All we knows for certain is that x must be divisible by 2

For example it COULD beat the case that x = (1)(2)(3) = 6
Since 6 is NOT divisible by 4, we can see that statement II is not necessarily true

Answer: D

Cheers,
Brent