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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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