I need to show that for x>=1 x^3-x^5 is divisible by 12.
Not really sure how to start! :-s
Many thanks in advance. x
The trick you often use for these type of questions is proving that (x+1) and (x) give the same remainder.
Easy example: 4^x is always divisible by 4. You take (x+1) which is: 4^(x+1) = 4*4^x. When dividing with 4 you can throw out the 4 and you're left with 4^x. Now if you prove that x=1 is divisible by four, you've also proven it for x+1 and x+2 etc.
Your problem is a little harder ofcourse, and sometimes it's easier to use (x+2) or (x+3) instead of (x+1), but you'll just have to try a little. Hope this helped, and if you still can't solve it I'll try to give you a start.
That's a silly way to write the problem.
Are they deliberately being annoying?
LetShow that is divisible by 12 for
Let's ignore the leading minus-sign.
With 3 consecutive integers, one of them is a multiple of 3.
. . Hence, is divisible by 3.
There are two cases: .
. . Therefore, is divisible by 12.