Use the Euclidean algorithm to determine the GCD(328, 132).  Show your work. Then express the GCD(328, 132) value you identify as a linear combination of 328 and 132. Show your work.

According to Fermat’s little theorem, what is the value of 3374(mod 11)?

The octal expansion of the decimal number 109 is ___________eight. Only type the digits; do not include the base.

The pet shop in Mochdewton sells only guinea pigs and hamsters. Each guinea pig costs twice as much as a hamster. A local elementary school purchased 5 guinea pigs and 3 hamsters. If the transaction had been for 3 guinea pigs and 5 hamsters, the cost would have been $10 less. What is the price of a guinea pig?  [GuineaPig] What is the price of a hamster?   [Hamster]   Adapted from “The Lady or the Tiger? and other logic puzzles” by Raymond M. Smullyan

The binary expansion of the decimal number 105 is ___________two. Only type the digits; do not include the base.

To complete the division algorithm equation, a = mq + r, using a = – 38 and m = 7, which of the following gives appropriate values for integers q and r, with r expressed as a non-negative integer between 0 and (m-1), inclusive.

For arbitrary integers a, b, and d, with d ≠ 0, if there are integers s and t such that d = as + bt, then d = GCD(a, b).

Determine which of these set identities are supported by the entries in the membership table given below.  There may be more than one or none. Select ‘True’ if the identity is supported by this given membership table; otherwise select ‘False’. [1]  (A – B) – C ⊈ (A – B) [2]  (A – C) – B ⊂ (A – C) [3]  (A – B) ⊄ (A – C) – B [4]  (A – C) ≠ (A – B) – C [5]  (A – C) ⊆ (A – B) [6]  (A – C) – B = (A – B) – C A B C A – C A – B (A – C) – B (A – B) – C 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 1 1 0 1 0 1 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0

Use the Euclidean algorithm to determine the GCD(286, 273).  Show your work. Then express the GCD(286, 273) value you identify as a linear combination of 286 and 273.  Show your work.

Prove the following statement using induction. “For all positive integers n, 3|(n3 + 2n).”  Use good proof technique.  Note:  To avoid the need for typing superscript exponents, you may use the expression ‘n^3’ to represent n3. Grading rubric:1 pt. State the basis step, then prove it.1 pt. State the inductive hypothesis.2 pt. Complete the proof of the inductive step.1 pt. State the final conclusion at the end of the proof.1 pt. Label each part: the basis step, inductive hypothesis, inductive step, and conclusion.