# Numbers (urgent)

• Aug 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
• Aug 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) \$\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\$)
• Aug 10th 2009, 04:20 AM
stapel