Exam 2 CMSC 203 Fall 2009
1. Circle T if the corresponding statement is True or F if it is False.
T F For any positive integer, n, GCD(n, 1) = 1.
T F Every positive integer is either prime or composite.
T F If a ≡ b mod p, then (a/p) = (b/p)
T F If a and b are positive integers, then a = b(a DIV b) + (a MOD b).
T F 1 + 3 + 3
2
+ 3
3
+ 3
4
+ ... + 3
33
= 3
34
− 1.
T F Algorithms with O(n
3
) are less efficient than those with O(3
n
).
T F Any statement validated by the Weak Principle of Mathematical Induction
cannot be validated by the Strong Principle of Mathematical Induction.
T F GCD(368, 60) = GCD(60, 8).
2. Find GCD( 2
3
5
2
7
4
11
1
13
0
17
5
19
0
23
1
29
1
31
2
, 2
4
5
1
7
0
11
2
13
4
17
4
19
3
23
1
29
0
31
3
)
3. List out the search intervals of the Binary Search algorithm to find 4 in the list:
3 4 6 9 13 18 21 34 55 72 83 85 92 104 111 133
4. List the next 5 terms of the sequence {a
n
} that follows the Fibonacci relation with the initial
conditions, a
0
= 2 and a
1
= 4.
5. Write out the Division Algorithm and trace its steps to calculate (44 MOD 12).
6. What set, S, is defined by the Inductive Definition:
1 ∈ S, and, if n ∈ S, then (5n) ∈ S.
7. Show n
10
is the Big-Oh of the algorithm with complexity:
(2n
3
+ 3n
2
)(n
7
+ 6n
5
) + [3n
5
+ 2n
4
].
8. Prove ONE of the TWO Theorems below using Mathematical Induction.
Theorem 1: For all Natural numbers n, .
Theorem 2: If a
0
= 10, a
1
= 20, and a
2
= 30, then a
n
= a
n−1
+ a
n−2
+ a
n−3
is a multiple of 10,
for all n > 2.
9. Prove ONE of the TWO Theorems below:
Theorem 1: If a, b, and c are Integers, with a = b + c, then GCD(a, b) = GCD(b, c).
Theorem 2: If a, b, c, d, and p are Integers with a ≡ b mod p, and c ≡ d mod p, then
(a + c) ≡ (b + d ) mod p.
10. Prove ONE of the TWO Theorems below by Contradiction or Contraposition.
Theorem 1: The set of Natural numbers is infinite.
Theorem 2: If n is an Integer and n
2
is even, then n is even.
7
i
i0=
n
∑
7
n1+
1Ð
6
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=
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