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Unformatted text preview: Questions for the exam 1 Theoretical questions 1. Formulate binomial theorem. Right the formula for ( x + y ) n and formula for the binomial coefficients. 2. For two positive integer numbers n,m define gcd ( n,m ) and prove that it exists. You can not use prime number factorization, you can use well ordering principle. 3. For two polynomials n ( x ) ,m ( x ) define gcd ( n ( x ) ,m ( x )) and prove that it exists. 4. Prove that if integers numbers a and c are relatively prime and c divides ab then c divides b . 5. Prove that if polynomials a ( x ) and c ( x ) are relatively prime and c ( x ) divides a ( x ) b ( x ) then c ( x ) divides b ( x ). 6. Prove that there are infinitely many prime numbers. 7. Prove that if a product of several numbers is divisible by a prime number p then one of them is divisible by p . 8. Prove that if a product of several polynomials is divisible by an irreducible polynomial p ( x ) then one of them is divisible by p ( x )....
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