HKD1710 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 (in other words, x is divisible by 3)
II. x is an integer multiple of 4 (in other words, x is divisible by 4)
III. x is an integer multiple of 6 (in other words, x is divisible by 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 1So, 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 trueAlso, if x is divisible by 3 and 2, then x is also divisible by 6.
So
statement III is trueWhat 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 trueAnswer: D
Cheers,
Brent
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