BTGmoderatorDC wrote:If k is an integer greater than 6, all of the following must be divisible by 3 EXCEPT
A. k(k + 3)(k - 1)
B. 3k^3
C. (k+1)(k+5)(k+6)
D. (k+2)(k-2)(k+3)
E. k(k+1)(k+2)
OA A
Source: GMAT Prep
Note that one among three consecutive integers must be divisible by 3. Let's see. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, ...
So, one among k(k+1)(k+2) is divisible by 3. Thus, for example, if (k+1) is divisible by 3, (k+1+3) = (k+4) is divisible by 3.
Let's analyze each option one by one.
A. k(k + 3)(k-1): It can be written as (k-1)k(k + 3)
We see that the above are not three consecutive terms. If (k-1) and k are not divisible by 3, then (k+1) and (k+1+3) = (k+4) must be divisible. Since the third term (k+3) lies between (k+1) and (k+4), we are sure that k(k + 3)(k-1) is NOT necessarily divisible by 3. Correct answer.
Though we got the answer, let's analyze others, too.
B. 3k^3: Certainly divisible as 3k^3 has 3 as a factor.
C. (k+1)(k+5)(k+6):
"¢ Say (k+1) as well as (k+2) is not divisible by 3, then (k+3) and (k+3+3) = (k+6) are divisible by 3, so, (k+1)(k+5)(k+6) is disible by 3.
"¢ Say (k+1) is not divisible by 3 but (k+2) is divisible by 3, then (k+2+3) = (k+5) is divisible by 3, so, (k+1)(k+5)(k+6) is disible by 3.
D. (k+2)(k-2)(k+3): Let write it as (k-2)(k+2)(k+3)
"¢ Say (k-2) as well as (k-1) is not divisible by 3, then k and (k+3) are divisible by 3, so, (k+2)(k-2)(k+3) is divisible by 3.
"¢ Say (k-2) is not divisible by 3 but (k-1) is divisible by 3, then (k-1) and (k-1+3) = (k+2) are divisible by 3, so, (k+2)(k-2)(k+3) is divisible by 3.
E. k(k+1)(k+2): This is a product of three consecutive integers, certainly divisible by 3.
The correct answer:
A
Hope this helps!
-Jay
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