If x is the product of three consecutive positive integers

Question

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

Gladiator59 · Accepted Answer

For any three consecutive positive integers, at least one of them must be even as every alternate number is even. On the other hand, at most two can be even.

Since every third number is divisible by 3, for any three numbers one has to be an integer multiple of 3.

Based on the two above facts - I and III are true.

Now, since every fourth number has to be an integer multiple of 4, based on our selection there might be either one or two even numbers in our list and hence integer multiple of 4 cannot be guaranteed.

Hence Option (D) is our choice.

Best,
Gladi

Arro44 · Answer

I would say the answer to this question is D.

Three consecutive positive integers can always be divided by 3.
Furthermore, we have the relevant conditions for a number divisible by 6 as well as there are always multiples of two and three involved.

In case my answer is wrong, please let me know where I went off the road.

Best regards,
Chris

Salsanousi · Answer

a * (a+1) * (a+2)

if a = 1 then 1 * 2 * 3 = 6 so divisible by 3 and 6

Answer choice D

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KSBGC · Answer

key points:

3 consecutive integers contain odd and even both types for sure. Even number is divisible by 2.

Another point about 3 consecutive integers is that their product is divisible by 3.

Any product that is divisible by both 2 and 3 is divisible by 6 too.

D is the correct answer.

Aamirso · Answer

Out of three consecutive integers:

- At least one is a multiple of 3.
- At least one is even.

Hence, 
I must be true.
II may be true.
III must be true.

I and III.

D is my answer.

Gmatprep550 · Answer

Not able to find why option 2 is not possible, Can anyone please advise.

dcummins · Answer

Test (1)(2)(3)=n
6 is not divisible by 4

BrentGMATPrepNow · Answer

----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

ThatDudeKnows · Answer

I. Every third positive integer is a multiple of 3. No matter by what we multiply that multiple of 3, we will always get a multiple of 3. We need I. B is out. 
II. A multiple of 4 requires two 2s. If our numbers go odd, even, odd, we must have at least one 2, but we aren't required to have two of them (for example: 1, 2, 3). We can't have II. C and E are out.
III. We know we have at least one even number and one multiple of 3 (they can be the same number), so we must end up with a multiple of 6. We need III. A is out.

Answer choice D.

Paras96 · Answer

Let's analyze each statement:

I. x is an integer multiple of 3:
To check if x is divisible by 3, we consider the product of three consecutive positive integers. One of these three consecutive integers must be divisible by 3 because every third positive integer is divisible by 3. Therefore, statement I is true.

II. x is an integer multiple of 4:
To check if x is divisible by 4, we need two consecutive even integers in the product. However, among three consecutive positive integers, there is always one odd integer, and no matter how you arrange them, you can't have two consecutive even integers. So, statement II is not necessarily true.

III. x is an integer multiple of 6:
To check if x is divisible by 6, we need two conditions met: the presence of a factor of 2 (even integer) and a factor of 3 (multiple of 3). In the product of three consecutive positive integers, we will always have an even integer because every other integer is even. Additionally, as stated in statement I, one of these integers must be divisible by 3. Therefore, statement III is true.

So, the correct answer is:
 (D) I and III only

bumpbot · Answer

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If x is the product of three consecutive positive integers

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If x is the product of three consecutive positive integers

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