Number properties

[JackH825]

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 ?

Please can you explain this? Taken from GMAT Official pratice Qs. I have read the explanations but can't get my head around it.

Thanks!

[Sage Pearce-Higgins]

Please quote the whole problem that you're discussing when posting here (including answer choices and the correct answer).

The basic tenet of probability is that we give a fraction between 0 and 1 with the denominator as 'total possible outcomes' and the numerator as 'desired outcomes'. In the simple case of rolling a die, we have the probability of rolling a "4" as 1 desired outcome divided by the total number of possible outcomes, thus 1/6.

In this case we see that we've got 96 possible outcomes for n, that's our denominator. Now it's up to us to decide how many of those possible outcomes give us a situation in which n(n+1)(n+2) is divisible by 8. If you're thinking of multiplying out those parentheses, don't! Divisibility problems are usually much more easily solved with factorized forms, i.e. something multiplied by something, as that shows some things that the number is divisible by.

If this sounds confusing, remember that examples can help us, so let's pick one. If n = 1, then n(n+1)(n+2) = 1x2x3 = 6, not divisible by 8. Go ahead an pick some more examples and see if you can spot a pattern before reading on...






Yes, you got it, if n is even, then n(n+1)(n+2) is divisible by 8. So that gives us 48 desired outcomes out of 96, so it looks like the answer is 1/2. But wait! It looks like if n is odd then n(n+1)(n+2) is not divisible by 8, and that would make sense because we have odd x even x odd, but there are some exceptions. What if the middle number was, say, 8? Clearly 7 x 8 x 9 is divisible by 8. So some odd values for n give us a desired outcome, meaning that the answer is a bit more than 1/2. That's enough to solve this problem.

It's great to go into more depth to understand why, but please be aware that what follows isn't necessary in order to solve the problem above. I encourage students to "think in prime factors" when dealing with divisibility; applying that to 8 means that we can rephrase 'is this number divisible by 8?' as 'does this number have 3 prime factors of "2"?'. It's also useful to notice that this problem is dealing with consecutive integers, i.e. we have 3 consecutive integers multiplied together.

If n is even, then we have even x odd x even. That means that we definitely have 2 prime factors of "2" in this multiplication. Also, be aware that every other multiple of 2 is a multiple of 4 (write out the multiples of 2 to see this), so that one of those even numbers is going to have 2 prime factors of "2" and the other is going to have 1 prime factor of "2". That makes 3 in total, so that the multiplication is divisible by 8.

If n is odd, then just a few examples of n(n+1)(n+2) are divisible by 8. Which ones? In this case there's just one even number (the middle one), so that would have to be a multiple of 8. How many multiples of 8 are there from 1 to 96? Simply divide by 8 to get 96 / 8 = 12.

Finally, let's add together all the "desired outcomes". We had 48 even values for n, and 12 values for n that give (n+1) as a multiple of 8. That makes the answer 60 / 96 = 5 / 8

[JackH825]

Brilliant, super helpful as always

[Sage Pearce-Higgins]

You're welcome.