Same sum, rewritten in an alternate font that might be easier to read

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Problem 1: Consider the function f : R —> R defined by f(x) = [xJ, where [ J denotes the "floor" function, sometimes also called the 'greatest integer' function. 

(A) What is the domain of f? 

(B) What is the codomain of 1?

(C) What is the range of f? 

(D) Is f one-to-one? If yes, explain why. If not, give a counterexample. 

Problem 2: Rewrite the sum below in sigma notation. Do not evaluate or simplify to a final answer. 

221 + 43! + 61! + 85! +106f +

Same sum, rewritten in an alternate font that might be easier to read: 

2^(2!) + 4^(3!) + 6"(4!) + 8^(50 + 161^(61) + 12A(7!) 

Problem 3: I nt I {a, r, t} and S = fl, a, x}. Compute T x S. 

Problem 1: Answer the following prompts with true or false. 

(A) 6 is a multiple of —2 

(B) 2 is divisible by 6 

(C) 3 1 6

(D) —316  

(E) 3 1 —6 

(F) 5 divides 20  

(G) 20 divides 5 

Problem 2: Find the values of q and r guaranteed by the Division Algorithm when —58627 is divided by 27. 

Problem 3: (A) There are 20 professors in a department. The department needs a four-person curriculum committee for next year. How many different committees are pos-sible? 

(B) How many 6 letter names (using the 26 letter English alphabet) have exactly two vowels (AEIOU)? 

(C) Person A and Person B are getting married! Person A has 5 close friends or family members in the wedding party. Person B also has 5 close friends or family members in the wedding party. The wedding party is lining up for pictures. In how many ways can the people be arranged if, from left to right, we have all of A's friends/family members, then Person A, then Person B, then all of B's friends/family members? 

Problem 4: Trace the execution of Euclid's Algorithm to compute gcd(58627, 27).

Problem 5: How many length 6 bit strings contain 110 as a substring? 

Problem 6: Suppose a and b are positive integers, n > 1 is an integer, and c is any integer. Prove that if a e b mod n then (a + c) e (b + c) mod n. 

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