Exam 2 CMSC 203 Fall 2010
1. Circle T if the corresponding statement is True or F if it is False.
T F GCD(0, 1) = 1.
T F Every integer can be factored as the product of primes.
T F If a โก b mod 5, then a and b are both multiples of 5.
T F If a and b are positive integers with a < b, then 0 < (b MOD a) < a.
T F 1 + 2 + 2
2
+ 2
3
+ 2
4
+ ... + 2
9999
= 2
10000
โ 1.
T F Algorithms with O(n) are less efficient than those with O(log n).
T F The Principle of Mathematical Induction is useful for discovering new
Mathematical theorems.
T F GCD(10,000,000, 9,999,998) = 2.
2. Find GCD( 2
3
5
2
7
4
11
1
17
5
23
1
29
1
31
2
, 2
4
5
1
11
2
13
4
17
4
19
3
23
1
31
3
)
3. List out the search intervals of the Binary Search algorithm to find 111 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
}, a
n
= a
n โ 1
+ 2a
n โ 2
with the initial
conditions, a
0
= 0 and a
1
= 1.
5. Trace the Euclidean Algorithm below to find GCD(52, 6).
PROCEDURE GCD(A,B: integers) Pass
TEMP = (A MOD B) A
WHILE (TEMP > 0)
A = B B
B = TEMP
TEMP = (A MOD B) TEMP
ENDWHILE
OUTPUT (B) (TEMP > 0)?
6. What set, S, is defined by the Inductive Definition: 2 โ S, and, if n โ S, then (n + 5) โ S.
7. Show n
12
is the Big-Oh of the algorithm with complexity: (2n
2
+ 5n
2
log n)(n
6
+ 2n
4
)(3n
2
+ n
3
).
8. Prove ONE of the TWO Theorems below using Mathematical Induction.
Theorem 1: For all Natural numbers n, .
Theorem 2: If a
0
= 5, a
1
= 10, and a
2
= 15, then a
n
= a
nโ1
+ 2a
nโ2
+ 3a
nโ3
is divisible by 5,
for all n > 3.
9. Prove ONE of the TWO Theorems below:
Theorem 1: For all Integers, n, if n
2
is even, then n
2
is divisible by 4.
Theorem 2: If a and b are Integers with a < b, then there is a Rational, q, such that a < q < b.
10. Prove ONE of the TWO Theorems below by Contradiction or Contraposition.
Theorem 1: The square root of an Irrational Number is Irrational.
Theorem 2: For all Integers, n > 2, if n is prime, then n is odd.
i
2
i0=
n
โ
n n 1+( ) 2n1+( )
6
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