enigma123 wrote:
Guys,
These type of problems are really becoming a pain for me. can you please tell me what to do and where I am going wrong?
Question: If x^3 - x=p, and x is ODD, is p divisible by 24?
My solution: Factor x to give me x(x-1)(x+1) = p. So when x is ODD then (x-1) (x+1) will be even (ODD+-ODD=Even). That means p is even.
Now 24 = 2^3 * 3. Until here I am fine. And after this I get stuck on most of these types of problems. What to look for after this to get an answer?
There, there.
Okay, you're off to a good start: factoring and breaking down expressions and numbers is an excellent habit. But the "playing around" with numbers cannot stop there. You need to take this a step further and think about two important points:
First, if x is an integer, (x - 1), x and (x + 1) are by definition consecutive integers. Among three consecutive integers, one of them must be a multiple of 3.
Second, if x is odd, (x - 1) and (x + 1) are both even. And since every second even number is a multiple of 4, one of the two has to be a multiple of 4. And since 4 times 2 = 8, a multiple of 4 times a multiple of 2 has to be a multiple of 8.
So, x(x - 1)(x + 1) has to have a multiple of 3 somewhere in there, and it has to have 2^3 multiplied in there as well. And since 3 and 2 are both prime and neither can be overlapped with prime factors, we conclude that p must be a multiple of (2^3) * 3.
That help?