Numbers (urgent)

• August 10th 2009, 01:26 AM
justanotherperson
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
• August 10th 2009, 02:51 AM
alexmahone
Quote:

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) $2^4=-1\ (mod\ 17)$

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

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

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

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

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

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

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

Now expand using binomial theorem. (Note that $62=31*2$)
• August 10th 2009, 04:20 AM
stapel