1 $If and are integers such that and, what can be concluded about and ?$

A.

B.

C.

or

D.

2 $What is the value of ?$

A.

2

B.

3

C.

5

D.

17

3 $Which of the following integers is prime?$

A.

1

B.

27

C.

29

D.

51

4 $The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be written uniquely as a product of:$

A.

Odd numbers

B.

Prime numbers

C.

Composite numbers

D.

Integers modulo

5 $Calculate .$

A.

6

B.

12

C.

24

D.

72

6 $If, then and are said to be:$

A.

Twin primes

B.

Relatively prime (or coprime)

C.

Composite pairs

D.

Congruent

7 $Given integers and, which formula correctly relates their GCD and LCM?$

A.

B.

C.

D.

8 $Using the Euclidean Algorithm, what is the first step to find ?$

A.

B.

C.

D.

9 $Bezout's Lemma states that if, then there exist integers and such that:$

A.

B.

C.

D.

10 $Which of the following linear congruences has a solution for ?$

A.

B.

C.

D.

11 $What is the multiplicative inverse of $3$ modulo $7$?$

A.

2

B.

3

C.

4

D.

5

12 $Find the least common multiple (LCM) of $4$ and $6$.$

A.

2

B.

12

C.

24

D.

10

13 $If and, which of the following is NOT necessarily true?$

A.

B.

C.

D.

14 $A linear congruence has a unique solution modulo if:$

A.

B.

C.

D.

is prime

15 $According to the Chinese Remainder Theorem, the system and has a unique solution modulo if:$

A.

B.

C.

D.

16 $Solve the system: and .$

A.

8

B.

13

C.

23

D.

5

17 $Fermat's Little Theorem states that if is a prime number and is an integer not divisible by, then:$

A.

B.

C.

D.

18 $Using Fermat's Little Theorem, what is ?$

A.

1

B.

2

C.

3

D.

6

19 $In a Caesar cipher with a shift of, what does the plaintext letter 'A' encrypt to?$

A.

C

B.

D

C.

X

D.

Z

20 $The decryption function for a Caesar cipher is:$

A.

B.

C.

D.

21 $An affine cipher encrypts using the function . Which condition must satisfy?$

A.

B.

must be even

C.

D.

22 $What is the inverse of $5$ modulo $26$?$

A.

5

B.

21

C.

1

D.

-5

23 $In the affine cipher, how do you decrypt a ciphertext ?$

A.

B.

C.

D.

24 $How many possible keys are there for a standard Caesar cipher (excluding the trivial shift of 0)?$

A.

25

B.

26

C.

1

D.

Infinite

25 $Which attack is the Caesar cipher most vulnerable to?$

A.

Brute force (Key exhaustion)

B.

Frequency analysis

C.

Both A and B

D.

None of the above

26 $Compute .$

A.

2

B.

-3

C.

3

D.

1

27 $If is an integer, what is ?$

A.

0

B.

1

C.

D.

28 $Which of the following pairs are relatively prime?$

A.

(14, 21)

B.

(15, 25)

C.

(9, 16)

D.

(12, 18)

29 $Find the value of if .$

A.

1

B.

2

C.

3

D.

4

30 $The congruence has:$

A.

One solution

B.

Five solutions

C.

No solution

D.

Infinite solutions

31 $Calculate .$

A.

1

B.

2

C.

102

D.

200

32 $Which of the following is equivalent to ?$

A.

B.

C.

for some integer

D.

33 $What is the Euclidean Algorithm used for?$

A.

Finding prime numbers

B.

Finding the Greatest Common Divisor (GCD)

C.

Encrypting messages

D.

Solving quadratic equations

34 $If is prime, what is the value of for any integer ?$

A.

0

B.

1

C.

D.

35 $In the context of the Chinese Remainder Theorem, finding involves computing and their inverses such that:$

A.

B.

C.

D.

36 $Which integer is its own multiplicative inverse modulo 5?$

A.

2

B.

3

C.

4

D.

0

37 $To solve, first divide the equation by to get:$

A.

B.

C.

D.

38 $In an affine cipher, the total number of possible keys is:$

A.

26

B.

312

C.

676

D.

12

39 $What is the remainder when $11$ is divided by $3$?$

A.

1

B.

2

C.

3

D.

0

40 $If, then the congruence has:$

A.

Exactly one solution

B.

No solution

C.

Multiple solutions

D.

Infinite solutions

41 $Which property allows us to write ?$

A.

Associativity

B.

Modular Multiplication Property

C.

Transitivity

D.

Commutativity

42 $Calculate .$

A.

1

B.

12

C.

35

D.

70

43 $Using the extended Euclidean algorithm, find integers such that .$

A.

B.

C.

D.

44 $The number of integers between 1 and 10 (inclusive) that are relatively prime to 10 is:$

A.

2

B.

3

C.

4

D.

5

45 $In cryptography, converting a message into a coded form is called:$

A.

Decryption

B.

Encryption

C.

Analysis

D.

Hashing

46 $If, which of the following is a valid value for ?$

A.

14

B.

16

C.

25

D.

-2

47 $Which of the following is a valid value for if ?$

A.

16

B.

14

C.

18

D.

20

48 $The set of integers forms a group under addition modulo . What is the identity element?$

A.

1

B.

0

C.

D.

49 $For the Caesar cipher, if the ciphertext is 'E' and the key is 2, the plaintext is:$

A.

C

B.

G

C.

B

D.

F

50 $Find$

A.

0

B.

1

C.

2

D.

3

51 $What is the smallest positive integer solution to and ?$

A.

2

B.

5

C.

7

D.

1

Unit 6 - Practice Quiz