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Review Problems, Exam 3 Math 2440 – Discrete Structures and Algorithms, I

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Review Problems, Exam 3 Math 2440 – Discrete Structures and Algorithms, I
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  Review Problems, Exam 3Math 2440 – Discrete Structures and Algorithms, I 1. Determine whether the integers in each of these sets are mutually relatively prime:(a) 8, 10, 12; (b) 15, 21, 28; (c) 8, 21, 25.2. Encrypt the message APPLES AND ORANGES using a shift cipher with  k  = 13.3. Find 210 div 17 and 210 mod 17.4. List all integers between -50 and 50 that are congruent to -2 modulo 30.5. Find the binary expansions of 235 and 1517.6. What is the decimal expansion of the number with(a) octal expansion (7016) 8 ?(b) binary expansion (1 1111) 2 ?7. Find gcd(35, 78), gcd(101, 203) and gcd(1000, 5040).8. Find the prime factorization of each of these integers: 729, 128, 44, 1002.9. Find a sequence of pseudorandom numbers that is generated using x n +1  = (4 x n  + 1) m od  7with seed  x 0  = 3.10. Decrypt the message HDW GLP VSP, that was encrypted using the Caesar cipher ( f  (  p ) =(  p  + 3) mod 26).11. Prove that2 − 2 · 7 + 2 · 7 2 + ··· + 2( − 7) n = 1 − ( − 7) n +1 4for every nonnegative integer n.12. Prove that for every positive integer n, we have1 · 2 + 2 · 3 + ··· +  n · ( n  + 1) =  n ( n  + 1)( n  + 2)3  . 13. Prove that any postage of n cents,  n ≥ 8, can be formed using just 3-cent and 5-cent stamps.14. Prove that 2 n  + 3 ≤ 2 n for  n ≥ 4.1
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