Number Prop DS

[espyn]

If x3 - x = n and x is a positive integer greater than 1, is n divisible by 8?

(1) When 3x is divided by 2, there is a remainder.

(2) x = 4y + 1, where y is an integer.

How should I solve this? Thanks in advance.

[Harish Dorai]

First simplify x3 - x (Read it as x-cubed - x) as x (x2 - 1) which is x (x + 1) (x - 1).
The above is a product of 3 consecutive numbers and we are asked whether this product is a multiple of 8. The product of 3 consecutive integers can be a multiple of 8, under the following conditions.

Condition 1) First integer is Odd, Second Integer is Even and also a multiple of 8 and Third Integer is Odd. Example: 7, 8, 9
Condition 2) First integer is Even, Second Integer is Odd and Third integer is Even. Example: 2,3,4 or 4,5,6

Now let us take the answer choices. We will start with Statement (2). It says x = 4y + 1 where y is an integer. This means x is 1 more than a multiple of 4. That means (x - 1) is a multiple of 4 and hence an even integer. So it satisfies Condition (2) mentioned above. So this is sufficient.

Statement (1) says that 3x divided by 2 yields a remainder. This means x is Odd. That means (x-1) is Even and (x+1) is Even. So again it satisfies Condition (2) above. So this statement is also sufficient.

So the answer is (D), I guess.

[RonPurewal]

Harish's explanation is wonderful, and entirely correct.

I only want to add a bit of clarification: If we have a string of three consecutive integers, the MIDDLE one of which is odd (the second case listed by Harish in his explanation), then ONE of the two even integers on the ends of the string is a multiple of 4 (because every other even integer is a multiple of 4). Therefore, the product of that integer and the other even integer will be a multiple of 4 x 2 = 8.

[CarynM701]

Hi Ron - I came to the correct answer a little differently on this problem. I factored the equation as stated x(x+1)(x-1). I restated the question as "Is x divisible by 8?" I did not notice that these were 3 consecutive integers and that X must be odd but I still came up with the correct answer D using this rephrased question. Was this by chance, or does this rephrase work as well? The way I worked this out is like this:

(1) 3x/2 has a remainder. Therefore, X must be odd and cannot be divisible by 8 - Sufficient
(2) x=4y+1. Therefore, 4Y must be even and any even+1 = Odd so X again must be odd and cannot by divisible by 8 - Sufficient

I just want to ensure that I didn't come up with the correct solution but chance and using this logic on a future problem would still work.

Thanks for you help

[tim]

You totally made two mistakes that canceled each other out. Notice that you even got a different answer to the question at the top of the page using your misinterpretation. It is safe to say that if you always get a yes using the correct method but your method always gives you a no, then you have done something wrong.

[RonPurewal]

Hi Ron - I came to the correct answer a little differently on this problem. I factored the equation as stated x(x+1)(x-1). I restated the question as "Is x divisible by 8?"

how did you get to the red thing? that's definitely not a valid re-phrasing of the question.
whether x is divisible by 8 and whether that big product is divisible by 8 are two VERY different questions! this should be clear, of course -- there'd be little point in this problem in the first place otherwise.