If N is a positive integer, is (N^3 - N) divisible by 4?


1) n = 2k + 1, where K is an integer.

2) n^2 + n is divisible by 6

We can rephrase the statement as such:

Is: n(n^2 - 1) divisible by 4?

Is N(N-1)(N+1) divisible by 4?

Is the product of three consecutive integers divisible by 4?

Final rephrasing:

Is N an odd integer or is N a multiple of 4?

Evaluate the statements:

1) n = 2k + 1, where K is an integer.

2K + 1 will give us an odd integer for N. (YES)

The problem I had was with plugging in 0 for K.
2(0) + 1 = 1 0x1x2 = 0 (Is 0 divisible by 4? This was my biggest problem because it was the only example that wasn't stated in the Number Properties book!)

Highlight line below for answer.
OA: A

Hi,

Thats a good point. I googled and found that 0 is divisible by any number.

(1) n = 2k + 1, which means n is an odd number.

Also (n^3 - n) = n(n-1)(n+1) = (n-1)n(n+1) = Multiplication of 3 consecutive numbers. If n is odd then (n-1) and (n+1) should be even, which means they should be multiple of 2 . Hence this will be sufficient.

Also, when k=0 => n=1 => n3 -n= 0 , which is divisible by 4


(2) n^2 + n divisible by 6, is clearly insufficient


Thanks.

yes, 0 is divisible by 4 - and, as the poster below has noted, by any other positive integer you care to divide by. 0 is divisible by every positive integer.

note the following:
the only way you will encounter this sort of query is if you plug in your own numbers. in other words, the official problems WILL NOT require you to decide the issue of whether 0 is divisible by n (for whatever n); they restrict the scope of divisibility problems strictly to positive divisors and positive dividends.

however, you should still know this fact, because, as you have seen here, you will often encounter "extra" questions like this as artifacts of plugging in your own numbers. therefore, even though the gmat won't test the concept directly, you may still have to rely on it to solve the problem because of your number plugging.

--

as long as we're at it, if you encounter "negative multiples" in your number plugging adventures, then yes, those are divisible too. for instance, -4 is divisible by 4, as are -8, -12, and the whole lot.

Hi Ron,

I noticed you only mentioned positive numbers...
But in general...zero is divisible by all non zero numbers right ???

thats what i assumed so far...got confused now..

correct.

however, you will never, ever, ever have to answer questions about divisibility by negative numbers. the gmat doesn't pose such questions - divisibility problems are strictly limited to divisibility by positive integers - and i can't even concoct a possible number plugging scenario in which that sort of situation would ever arise.
this is why the original comment was restricted to positive integers, and, more importantly, why your thinking about the situation should also be restricted to such numbers.

still, if you're wondering out of pure curiosity, then, yes, 0 is divisible by negative integers, too. in fact, the rules for divisibility by negative integers are precisely the same as those for divisibility by positive integers, because sign turns out not to matter (e.g., all numbers divisible by 4 are also divisible by -4, and vice versa).