If n is an odd natural number, then factors intoa) If N is a odd natural number, show that a+b divides (a^n)+(b^n)
Hello, i want to share some problems that i found in the internet, but i cannot verify the solutions and others i dont know how to solve them,
a) If N is a odd natural number, show that a+b divides (a^n)+(b^n)
b) There exist a natural number N such that 1955 divides n^2+n+1 ?
c) Find the remainder when 4444^4444 is divided by 9.
d) p & q are different primes, show that
1) p^q + q^p i is congruent to p+q (mod pq)
2) (p^q + q^p)/(pq) is even, if p & q is not equal to 2.
a step by step solution will be great, please.. thank you!
there is no natural number satisfying the condition.
since 5 divides 1955, but (mod 5) has no solution, we obtain that there is no natural number such that 1955 divides .
for(d)(1): applying the fermat's little theorem, we get pq divides both .
for(2), that is impossible according to (1).