MA74 Midterm 3 - I November 21, 2006 Instructor: Patrick BarrowName (print):Signature: → ‘I’m not gonna cheat.’0. Define the phrase ‘n ∈ N is prime’, fully defining any auxiliary terms used (such as ‘divisor’). Thenstate carefully the Fundamental Theorem of Arithmetic (= existence and uniqueness of prime factorization).1. Let x be a real numb er not equal to 1, and let n ∈ N be at least 2. Show by induction thatxn− 1x − 1= 1 + x + ... + xn−12. Show that no integer of the form 8k + 3 is the square of an integer.3. Suppose n ∈ N is NOT prime. Show that 2n− 1 is not prime.Hint: The formula you proved in the first problem is relevant here. But it will not help to let x = 2. Soyou need to use the hypothesis that n is not prime.4. Show that if f : N → N is bijective and increasing, then f is the identity function.Hint: At first, this statement seems ‘obvious’, but it is generally hard to prove such things. Usually, thismeans that the Well - Ordering Principle will help to get you started. So assume that f is not the identity,and define a set in a clever fashion...like the set of all n for which f (n) 6= n.5. Let n ∈ N. Show that gcd(n, 2n + 1) = 1.Hint: If you have done your homework, then this is the easiest problem on the test, and conversely.Bonus: What upper division math courses (if any) do you plan to take next
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