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If n is the product of 3 consecutive integers, which of the following

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If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 17 Nov 2016, 03:25
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) If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

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

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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 17 Nov 2016, 05:39
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MathRevolution wrote:
If n is the product of 3 consecutive integers, which of the following must be true about n?

I. n is a multiple of 2
II. n is a multiple of 3
III. n is a multiple of 4

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


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
NOTE: the product may be divisible by other numbers as well, but these divisors are guaranteed.

In this question, n is the product of 3 consecutive integers.
So, according to the rule, n must be divisible by 3, 2 and 1
So, we already know that statements I and II must be true.

Do we need to check statement III? No.
Notice that NONE of the answer choices include all 3 statements. Since we've already concluded that statements I and II are true, the correct answer is D.

HOWEVER, if you want to check statement III, notice that the product of 1, 2 and 3 (3 consecutive integers) is 6, and 6 is not divisible by 4

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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 17 Nov 2016, 07:51
MathRevolution wrote:
If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

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


There is a property which states -

Quote:
Product of three numbers is always divisible by 6


So, n will be a multiple of 2 and 3

Answer will be (D)

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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 21 Nov 2016, 01:02
==> The product of 3 consecutive integers always become the multiple of 6, because the product always contains 3 and 2. Thus, in this question, it always becomes the multiple of 6 that contain 3 and 2, I and II are the answer. Therefore, the answer is D. III does not work because it becomes 1*2*3*=6, hence it cannot be the multiple of 4.

Answer: D
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If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 21 Nov 2016, 01:49
MathRevolution wrote:
) If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

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


Among 3 consecutive integers, there is always at least one even integers, so n is divisible by 2. (I) is true

Among 3 consecutive integers, there is always only one integers divisible by 3, so n is divisible by 3. (II) is true.

3 consecutive integers could be even-odd-even or odd-even-odd.
If they are even-odd-even, n is divisible by 4.
If they are odd-even-odd, n is not divisible by 4. For example, \(n=1 \times 2 \times 3 =6\) is not divisible by 4.
Hence, (III) is not true.

Thus, the answer is D.
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 15 Jan 2017, 23:42
Product of n consecutive integers is always divisible by n!
So n will be of the form 6k for some integer k.
Clearly it will always be divisible by 2 and 3 but may/may not be divisible by 4.

Hence D.

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If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 22 Mar 2017, 01:52
If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

A. I only
B. II only
C. III only
D. I and II
E. II and III
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 23 Mar 2017, 15:57
MathRevolution wrote:
If n is the product of 3 consecutive integers, which of the following must be true?

I. a multiple of 2 II. a multiple of 3 III. a multiple of 4

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


Since n is the product of 3 consecutive integers, n is divisible by 3! = 6. Since 2 and 3 are factors of 6, Roman numerals I and II must be true.

Answer: D
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 23 Mar 2017, 18:43
can the integers be -1,0 and 1? in that case nothing holds true. I think it should have mentioned n not equal to 0.
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 24 Mar 2017, 02:56
==> If n is the product of 3 consecutive integers, it is always even and has 3, so it is always a multiple of 6. Thus, I and II is correct and for III, since n=1*2*3=6 is not a multiple of 4, hence it is incorrect.

The answer is D.
Answer: D
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 09 Apr 2017, 08:56
In three consecutive nos. one number (at least) will be 2 or a multiple of 2 thus making the product even. Thus divisible by 2.

The product of three consecutive nos. is always divisible by 3.
For example, 4*5*6 (6 is a multiple of 3)
7*8*9 (9 is a multiple of 3)
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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New post 11 Apr 2017, 18:40
Something you all need to know, an integer multiplied by the second and third consecutive will always be divisible by 2 and 3.
For an example
1 x 2 x 3.
There is no need to do any working out here. We all know that it is divisible by 2 and 3.
Therefore, the answer is D
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Re: If n is the product of 3 consecutive integers, which of the following  [#permalink]

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Re: If n is the product of 3 consecutive integers, which of the following   [#permalink] 03 Feb 2019, 21:13
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