CS 346 Cryptography
FALL 2010
Background Quiz
SOLUTIONS
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1. What is the numeric value of log
2
(8
50
)?
log
2
(8
50
) = 50
·
log
2
8 = 50
·
log
2
2
3
= 50
·
3 = 150.
2. Give a bigO estimate for each of the following functions:
A. 2
n
3
+
n
2
log
n
=
O
(
n
3
)
B. (
n
2
+ 8)(
n
+ 1) =
O
(
n
3
)
C.
n
! + 2
n
=
O
(
n
!)
3. A fair die is rolled 10 times, independently.
What is the probability
that a 6 appears at least once?
If a fair die is rolled once, the probability that a 6 does not appear is
5
6
.
If a fair die is rolled 10 times, the probability that a 6 does not appear
even once is (
5
6
)
10
.
So if a fair die is rolled 10 times, the probability that a 6 appears at
least once is 1

(
5
6
)
10
.
4. An integer is chosen at random from the first 300 positive integers.
What is the probability that the integer chosen is divisible by 2 or by
3?
Let
E
denote the event that the integer chosen is divisible by 2, and
let
F
denote the event that the integer chosen is divisible by 3. Then
P
(
E
) = 150
/
300 = 1
/
2 and
P
(
F
) = 100
/
300 = 1
/
3. We are looking
for the probability
P
(
E
∪
F
).
We know that
P
(
E
∪
F
) =
P
(
E
) +
P
(
F
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 Fall '10
 GAL
 Prime number, Greatest common divisor, Divisor, diﬀerent binary strings

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