PROBABILTY

[phelps]

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

CORRECT ANSWER

D

[Tani]

your three integers are consecutive so there are two basic possibilities

if n is even, both n and n+2 are even. Also, one of them must be divisible by 4, so the product will be divisible by 8. That happens 48 times.

if n is odd, you have 2 odd numbers and one even number. For the product to be divisible by 8,the one even number (the middle one) must be divisible by 8. that will happen if n = 7,15,23,31,39,47,55,63,71,79,87 or 95 ( 12 times)

Total = (48+12)/96 = 60/96 = 5/8

[havok]

your three integers are consecutive so there are two basic possibilities

if n is even, both n and n+2 are even. Also, one of them must be divisible by 4, so the product will be divisible by 8. That happens 48 times.

if n is odd, you have 2 odd numbers and one even number. For the product to be divisible by 8,the one even number (the middle one) must be divisible by 8. that will happen if n = 7,15,23,31,39,47,55,63,71,79,87 or 95 ( 12 times)

Total = (48+12)/96 = 60/96 = 5/8

Wow, good answer. I got 1/2 after thinking it only applied to even numbers. Didn't think to have other options where the number has 8 in it.

[venmic]

how do you get 48 times in the first instance and 12 in the next
other than counting the numbers what is the approach

your three integers are consecutive so there are two basic possibilities

if n is even, both n and n+2 are even. Also, one of them must be divisible by 4, so the product will be divisible by 8. That happens 48 times.

if n is odd, you have 2 odd numbers and one even number. For the product to be divisible by 8,the one even number (the middle one) must be divisible by 8. that will happen if n = 7,15,23,31,39,47,55,63,71,79,87 or 95 ( 12 times)

Total = (48+12)/96 = 60/96 = 5/8

[Tani]

The first list refers to groups of three consecutive numbers, each of which starts with an even number. There are 48 even number from 1-96 inclusive (simply divide 96 by 2).

For the second list, you have to have the middle number be divisible by 8. Dividing 96 by 8 you can see there are twelve numbers between 1 and 96 inclusive that are divisible by 8.