1. ## Numbers (urgent)

a) determine the remainder when 2^2009+1 is divided by 17

b) prove that 30^99+61^100 is divisible by 31

c) it is known that the numbers p and 8p^2+1 are primes. Find p

thnx a million
justanotherperson

PS: if u can include working out that would be great. XD

2. Originally Posted by justanotherperson
a) determine the remainder when 2^2009+1 is divided by 17

b) prove that 30^99+61^100 is divisible by 31

c) it is known that the numbers p and 8p^2+1 are primes. Find p

thnx a million
justanotherperson

PS: if u can include working out that would be great. XD
a) $\displaystyle 2^4=-1\ (mod\ 17)$

$\displaystyle (2^4)^{502}=(-1)^{502}\ (mod\ 17)$

$\displaystyle 2^{2008}=1\ (mod\ 17)$

$\displaystyle 2^{2009}=2\ (mod\ 17)$

$\displaystyle 2^{2009}+1=3\ (mod\ 17)$

The remainder when $\displaystyle 2^{2009}+1$ is divided by $\displaystyle 17$ is $\displaystyle 3$.

b) $\displaystyle 30^{99}+61^{100}$

$\displaystyle =(31-1)^{99}+(62-1)^{100}$

Now expand using binomial theorem. (Note that $\displaystyle 62=31*2$)