Question 1133818
<br>
We want to know when the expression<br>
{{{(C^3-C)/12}}}<br>
is an integer.<br>
{{{C^3-C = C(C^2-1) = C(C+1)(C-1) = (C-1)(C)(C+1)}}}<br>
So the expression is the product of three consecutive integers; we need to find the conditions under which the product of three consecutive integers is or is not divisible by 12.<br>
{{{12 = (2^2)(3)}}}<br>
So the product of three consecutive integers will be divisible by 12 if it contains two factors of 2 and one factor of 3.<br>
Every set of three consecutive integers contains exactly one which contains a factor of 3.  So we need to determine when the product of three consecutive integers contains two factors of 2.<br>
(1) If C is odd, then both C-1 and C+1 are even, so the product contains two factors of 2.
(2) If C is a multiple of 4, then that factor alone contains two factors of 2.
(3) If C is even but not a multiple of 4, then C-1 and C+1 are both odd; the product will contain only one factor of 2.<br>
So only 1 out of every 4 consecutive values of C will yield a product that is NOT divisible by 12.  So 3 out of every 4 WILL yield a product that is divisible by 12.<br>
There are 80 integers from 20 to 99 inclusive; since that number is a multiple of 4, we know that exactly 3/4 of them will yield a product that is divisible by 12.<br>
ANSWER: P(C^3-C is divisible by 12) = 3/4