alex.gellatly wrote:If an integer n is to be chosen at random from the integers 1 to 96, inclusive, what is the probability that n(n+1)(n+2) will be divisible by 8?
A. 1/4
B. 3/8
C. 1/2
D. 5/8
E. 3/4
Thanks in advanced
Case 1: n is even
Let us assume that n = 2k, for any integer k.
Then n(n + 1)(n + 2) = 2k(2k + 1)(2k + 2) = 4k(2k + 1)(k + 1)
Now either k or k + 1 will be even, so 8 will be a multiple of n(n + 1)(n + 2).
Number of even integers between 1 ans 96, inclusive = {(96 - 2)/2} + 1 = 48
Case 2: If n + 1 is divisible by 8
n + 1 = 8a, where a ≥ 1
n = 8a - 1
8a - 1 ≤ 96
8a ≤ 97
a ≤ 12.1 implies 12 integers.
Also, when n and n + 2 are even, n + 1 will be odd, and when n + 1 is divisible by 8, then n and n + 2 will be odd. This means, there is no repetition.
Total integers = 48 + 12 = 60
Therefore, probability that n(n + 1)(n + 2) will be divisible by 8 = 60/96 = [spoiler]5/8[/spoiler]
The correct answer is
D.
