math 311w midterm1 - Questions for the exam 1 Theoretical questions 1 Formulate binomial theorem Right the formula for(x y)n and formula for the

math 311w midterm1 - Questions for the exam 1 Theoretical...

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Questions for the exam 1 Theoretical questions 1.Formulate binomial theorem. Right the formula for (x+y)nand formula forthe binomial coefficients.2.For two positive integer numbersn, mdefinegcd(n, m) and prove that itexists. You can not use prime number factorization, you can use well orderingprinciple.3.For two polynomialsn(x), m(x) definegcd(n(x), m(x)) and prove that itexists.4.Prove that if integers numbersaandcare relatively prime andcdividesabthencdividesb.5.Prove that if polynomialsa(x) andc(x) are relatively prime andc(x) dividesa(x)b(x) thenc(x) dividesb(x).6.Prove that there are infinitely many prime numbers.7.Prove that if a product of several numbers is divisible by a prime numberpthen one of them is divisible byp.8.Prove that if a product of several polynomials is divisible by an irreduciblepolynomialp(x) then one of them is divisible byp(x).9.a)Prove that any positive integer number could be written as a product ofprime numbersn=p1p2· · ·pk

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