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

If n is even, then n = 2k for integers 1 ≤ k ≤ 48, and

n(n - 1)(n - 2) = 2k(2k - 1)(2k - 2) = 4k(k - 1)(2k - 1)

and this product is clearly divisible by 4. k(k - 1) is the product of two consecutive integers, and one of these must be even, so it follows that n(n - 1)(n - 2) is always divisible by 8. So

Pr[n(n - 1)(n - 2) is divisible by 8 AND n is even]

= Pr[n(n - 1)(n - 2) is divisible by 8 || n is even] * Pr[n is even]

= 1 * 1/2

= 1/2

(where I write Pr[A || B] to denote the conditional probability of A given B)

If n is odd, then n = 2k - 1 for integers 1 ≤ k ≤ 48, and

n(n - 1)(n - 2) = (2k - 1)(2k - 2)(2k - 3) = 2(k - 1)(2k - 1)(2k - 3)

This is clearly divisible by 2. We then need to find out when this is divisible by 4. We have

(k - 1)(2k - 1)(2k - 3) = 4k^3 - 12k^2 + 11k - 3

and we have divisibility by 4 if 4 divides 11k - 3. We have

11k - 3 = 0 mod 4

==>  11k = 3 mod 4

==>  33k = 3^2 = 1 mod 4

==>  (32 + 1)k = 1 mod 4

==>  k = 1 mod 4

That is, 4 divides 11k - 3 if k = 4r + 1 for integers 0 ≤ r ≤ 11. This means there are 12 integers r (and hence 12 integers k) for which 4 divides 11k - 3, which in turn means there are 12 integers n for which n(n - 1)(n - 2) is divisible 8. So

Pr[n(n - 1)(n - 2) is divisible by 8 AND n is odd]

= Pr[n(n - 1)(n - 2) is divisible by 8 || n is odd] * Pr[n is odd]

= 12/48 * 1/2

= 1/8

The cases that n is even or odd are mutually exclusive, so

Pr[n(n - 1)(n - 2) is divisible by 8 = 1/2 + 1/8 = 5/8

a net is a diagram that shows a “flattened” view of a solid. in a net, each face and base is shown with all of its dimensions. a net can also serve as a pattern to build a three-dimensional solid.

a pyramid is a three-dimensional object with a base that is a polygon and triangular faces that meet at one vertex.

" the net of a triangular pyramid consists of four triangles. the base here we are given of the pyramid is a equilateral   triangle, and the lateral faces are also triangles "