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If x is the product of three consecutive positive integers,
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18 Aug 2020, 09:39
18 Aug 2020, 09:39
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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
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
Re: If x is the product of three consecutive positive integers,
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18 Aug 2020, 09:40
18 Aug 2020, 09:40
Expert Reply
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Re: If x is the product of three consecutive positive integers,
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18 Aug 2020, 11:39
18 Aug 2020, 11:39
Carcass wrote:
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
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
----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
