1) Use the Euclidean algorithm to find the GCD or 1326 and 2940. Please show each stage of the process.

2) Let f(1) = p2 and f(n+ 1) = p1 + f(n). Prove, by induction, that f(n) is irrational for all positive integers n. You may use, without having to prove it, that p2 is irrational.

3) In class we gave the following definition of divisibility: For integers a, b we say that a divides b if there is an integer n such that an = b. Use this definition to prove that if a divides b and c divides d then ac divides bd. Please explicitly state any laws of arithmetic you use in the proof (so if you use the commutative law of addition, for example, then say that you are using the commutative law of addition)

4) Let [a, b] and [c, d] be any two closed intervals in the real numbers (for example [3,7] and [22,94]) Prove that [a, b] and [c, d] have the same cardinality by finding an explicit bijection between them.

5) a) Convert 267 into base 7 b) Write out a multiplication table for if numbers were written base 6 (so your table just needs to go to 5 by 5). Then use the table to multiply 43562146 Then convert the factors and the product to base 10, and show that your work is correct. Please show the work.

6) Prove, by induction, that for all n 1, 14+25+...+(11n+3) = n(11n+17)