[email protected] wrote:For some integer q, q² - 5 is divisible by all of the following EXCEPT
(A) 29
(B) 30
(C) 31
(D) 38
(E) 41
There's a nice rule that goes something like this:
If A is divisible by k, but B is not divisible by k, then A+B is NOT divisible by k, and A-B is NOT divisible by k.
There's another rule that says, "
If there are n consecutive integers, then 1 of them is divisible by n"
Okay, now onto the question.
IMPORTANT: Notice that only
answer choice B is divisible by 3. We'll use this fact in our solution.
Rewrite the given expression:
q² - 5 = q² - 1 - 4 = (q - 1)(q + 1) - 4
NOTICE that (q - 1), q and (q + 1) are 3 consecutive integers, so one of them must be divisible by 3 (from the red rule above). So, let's consider 3 possible cases:
(q - 1) is divisible by 3
So, (q - 1)(q + 1) must also be divisible by 3
Since 4 is NOT divisible by 3, we know that (q - 1)(q + 1) - 4 is NOT divisible by 3 (from the green rule above)
If (q - 1)(q + 1) - 4 is not divisible by 3, it CANNOT be divisible by 30
q is divisible by 3
So, q² must also be divisible by 3
Since 5 is NOT divisible by 3, we know that q² - 5 is NOT divisible by 3 (from the green rule above)
If q² - 5 is not divisible by 3, then it CANNOT be divisible by 30
(q + 1) is divisible by 3
So, (q - 1)(q + 1) must also be divisible by 3
Since 4 is NOT divisible by 3, we know that (q - 1)(q + 1) - 4 is NOT divisible by 3 (from the green rule above)
If (q - 1)(q + 1) - 4 is not divisible by 3, it CANNOT be divisible by 30
Now that we've examined all 3 possible cases, we can conclude that q² - 5 CANNOT be divisible by
30
Answer =
B
Cheers,
Brent
