Please show all work to support your answer

1. Let = {, , , , 1, 2, 3,{4,5}} which of the following is true [careful]?

A. {} B. C. 4 D. {1,2}

2. One of the following is a subset of = {1,2,3,{4,5,6},{9,10}}. Which one is it?

A. {4,5,6} B. {1,3,{4,5,6}} C. {9,10} D. 1

3. Let = 82 and = 16. Find and 0 < such that = + .

A. = 2 ; = 5 B. = 6 ; = 1 C. = 5 ; = 2 D. = 5 ; = 3

4. Which of the following is the prime factorization of 735?

A. 3 5 7²B. 3 5² 7 C. 3³ 5 7 D. 3² 5 7²

5. Which of the following numbers appears in the list of divisors of 735?

A. 30 B. 45 C. 147 D. 255

6. Which of the following represents 206 in the octal system (base 8 system)?

A. (316)₈B. (330)₈C. (246)₈D. (260)₈

7. Which of the following represents 35 in the binary system (base 2 system)?

A. (10011)₂B. (111011)₂C.(100001)₂D. (100011)₂

8. Please support your answer with clear and detailed arguments (as needed). Please show all your work. Note: () is the power set of and || is its cardinality.

If = {3,7,2}, find () and |()|

() =_________________________________________________________

|()| = _________

9. Let ₁ = 1, ₂ = 3 and _n = _n-1 + 2_n-2 for = 3, 4, 5, ... . Find ₅.

10. Let = {, , , , , , } be the universal set and = {, , } and = {, , } be subsets of . Compute () where _x is the indicator or characteristic function of set .

11. Find the greatest common divisor and least common multiple of the numbers 12345 and 67890. If is the greatest common divisor, find and such that = + .

GCD _____________

LCM _____________

= __________