Division, Primes, Divisors & Powers: Solve Them All!

[AlexHall]

1) Find the remainder of the division of 15! with 17

2) If (n^2)+2 prime show that 3 divides n

3)If p the smallest divisor for n show that there exist integers a and b such that an+b(p-1)=1

4) For every n>1 show that n does not divide (2^n)-1

Any help?

 

[Dick]

You really have to show some kind of effort (even if it's unsuccessful) before people can help. And break different questions up into different threads. But I'll give you a free hint for the first one. The integers mod 17 are a group under multiplication. Or just use Wilson's theorem. So what is 16! mod 17?

 

[AlexHall]

Thanks

16!=-1(mod17)
16!=15!.16=-1(mod17)

16=-1(mod17)

15!=1mod17

I don't know how to start the other problems. Any tips?

 

[Dick]

TRY one. Make an attempt. Do anything.