Hello, Just!
We have: .b) Prove that is divisible by 31
Expand the binomial: .
We have: .
. . . . . . .
The first group is: . . . . a multiple of 31
Hence: .
. . Therefore, is divisible by 31.
hi, can some one answer these and tell me hoe u got them?
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 numbers p and 8p^2+1 are primes. Find p
thnx so much u lifesaver
justanotherperson
Well Soroban helped you with the hardest one but here's some help on a and c) as well if you need it.
a) Observe that 2^8 = (15*17+1). That means you can just take 2009 mod 8 to find the power of 2 you need to calculate.
c) Here's a very big hint. What is 8p^2+1 mod 3 if p leaves a remainder of either 1 or 2 when divided by 3? You'll soon see a pattern if you try a few primes, and you should be able to prove it's true for all primes of either of those forms.