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# MATH HL portfolio(divisibility)

Hi
I got this IA 2 days ago and im current working HARD on it but i don't seem to get along well with question 3 and 4.I know some of you guys have already finished it so i need some help with this.

1. Factorize the expression P(n)=n^x-n for x = 2,3,4,5. Determine if the expression is always divisible by the corresponding x. If divisible use mathematical induction to prove your result by showing whether P(k+1)-P(k) is always divisible by x. Using appropriate technology, explore more cases and make a conjecture for when n^x - n is divisible by x.

2. Explain how to obtain the entries in Pascal's triangle...State the relationship between the expression P(k+1)-P(k) and Pascal's traingle. Reconsider your conjecture.

Write an expression for the xth row of the Pascal's Triangle. You will have noticed that (x r) = k, k is a natural number. Determine when k is a multiple of x.

3. Make conclusions regarding the last result in part 2 and the form of proof by inductiton used in this assignment. Refine your conjecture if neccessary, and prove it.

4. State the converse of your conjecture. Describe how you woul prove whether or not the converse holds.

Ok so for the second question,my hypothesis is that k is a multiple of x(when x is prime) then the expression divides its correspond x.But i done get this part"State the relationship between the expression P(k+1)-P(k) and Pascal's traingle".This is pretty much the same problem in question 3.

Thanks

One more small question:i researched about this and found that this is called "Fermat's lilttle theorem" but it is all about modular arithmetic which i have no idea of.Did anyone acttually used Fermat's theorem in theirs work

Thanks again

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