(Solved): Do number 2 PLEASE SUBMIT YOUR PAPER SCANNED TO PDF 1. a) Prove that every integer can be written a ...

PLEASE SUBMIT YOUR PAPER SCANNED TO PDF 1. a) Prove that every integer can be written a sum of powers of 2 . b) Write \( 14,17,27 \) as a sum of powers of 2 2. Let \( n \) be an integer \( >1 \) and \( N=n !+1 \) a) Prove that \( N \) has a prime divisor \( >n \) b) Use a) to prove that there are infinitely many prime numbers 3. a) Suppose \( a \equiv 1(\bmod 4) \) and \( b \equiv 1(\bmod 4) \), prove that \( a b \equiv 1(\bmod 4) \). b) If \( a \equiv 3(\bmod 4) \) and \( b \equiv 3(\bmod 4) \) what can you say about \( a b \) ? Justify your answer. 4. Prove that there are infinitely many primes congruent to 5 modulo 6. Make your proof clear and concise.