1 $If and are integers with, we say that divides (denoted) if there exists an integer such that:$

A.

B.

C.

D.

2 $Which of the following properties of divisibility is false for integers ?$

A.

If and, then .

B.

If and, then .

C.

If, then .

D.

If, then or .

3 $What is the Greatest Common Divisor (GCD) of 24 and 36?$

A.

6

B.

12

C.

18

D.

24

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.

Primes

C.

Composite numbers

D.

Squares

5 $For any two positive integers and, which of the following relationships holds true involving their GCD and LCM?$

A.

B.

C.

D.

6 $What is the result of ?$

A.

2

B.

3

C.

1

D.

7 $Two integers and are said to be congruent modulo (written) if:$

A.

B.

C.

and

D.

8 $Calculate the Least Common Multiple (LCM) of 4 and 6.$

A.

12

B.

24

C.

10

D.

2

9 $The Euclidean Algorithm is a method used to find:$

A.

The prime factorization of a number

B.

The Greatest Common Divisor (GCD) of two numbers

C.

The Least Common Multiple (LCM) of two numbers

D.

The next prime number

10 $According to Bezout's Lemma, for nonzero integers and, there exist integers and such that:$

A.

B.

C.

D.

11 $What is the value of ?$

A.

-1

B.

1

C.

2

D.

12 $Which of the following is true about prime numbers?$

A.

1 is a prime number.

B.

2 is the only even prime number.

C.

All odd numbers are prime.

D.

There are finitely many prime numbers.

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

A.

B.

C.

for integer

D.

14 $Find the multiplicative inverse of $3$ modulo $7$.$

A.

2

B.

3

C.

4

D.

5

15 $The multiplicative inverse of modulo exists if and only if:$

A.

divides

B.

divides

C.

D.

and are both prime

16 $Solve the linear congruence . How many distinct solutions are there modulo 6?$

A.

B.

1

C.

2

D.

6

17 $What is the condition for the linear congruence to have a solution?$

A.

B.

divides

C.

divides

D.

divides

18 $In the Euclidean algorithm, if and, what is the first remainder calculated?$

A.

4

B.

6

C.

2

D.

16

19 $Fermat's Little Theorem states that if is a prime number and is an integer not divisible by, then:$

A.

B.

C.

D.

20 $Using Fermat's Little Theorem, what is ?$

A.

1

B.

3

C.

6

D.

21 $The Chinese Remainder Theorem guarantees a unique solution modulo for a system of congruences if the moduli are:$

A.

All prime

B.

All equal

C.

Pairwise coprime

D.

Even numbers

22 $In a Caesar cipher with a shift of, the plaintext letter 'C' is encrypted as:$

A.

A

B.

E

C.

F

D.

Z

23 $The Caesar cipher is a special case of which type of cipher?$

A.

Affine Cipher

B.

RSA

C.

ElGamal

D.

Vigenere Cipher (with variable key)

24 $The decryption function for the Caesar cipher is:$

A.

B.

C.

D.

25 $An Affine cipher encrypts using the function . Which condition must satisfy for the cipher to be valid (invertible)?$

A.

B.

must be even

C.

D.

must be prime

26 $How many possible keys exist for the 'additive' part () of an Affine cipher over the English alphabet?$

A.

12

B.

25

C.

26

D.

Infinite

27 $In the Affine cipher, what is the decryption key for ?$

A.

-5

B.

21

C.

5

D.

1

28 $Decrypt the letter 'D' (value 3) using Caesar cipher with key .$

A.

G

B.

A

C.

H

D.

B

29 $Which integer is a solution to the system: and ?$

A.

4

B.

5

C.

2

D.

3

30 $What is ?$

A.

B.

1

C.

5

D.

Undefined

31 $If is a prime, how many solutions does have?$

A.

Exactly 1

B.

Exactly 2 (if)

C.

D.

32 $Which of the following numbers is prime?$

A.

21

B.

27

C.

29

D.

33

33 $Compute the modular inverse of 2 modulo 9.$

A.

4

B.

5

C.

2

D.

No inverse

34 $For the Affine Cipher, the decryption formula is . If encrypted ('K'),,, find the plaintext value .$

A.

17

B.

6

C.

11

D.

9

35 $Which of the following pairs are relatively prime (coprime)?$

A.

14 and 21

B.

15 and 25

C.

9 and 16

D.

8 and 12

36 $What is the value of where is Euler's totient function?$

A.

7

B.

6

C.

1

D.

5

37 $To use the Chinese Remainder Theorem to solve, let . The solution involves finding and which is the modular inverse of:$

A.

modulo

B.

modulo

C.

modulo

D.

modulo

38 $What is the remainder when is divided by 7?$

A.

1

B.

2

C.

4

D.

3

39 $In modular arithmetic, is equal to:$

A.

B.

C.

D.

40 $Which algorithm allows us to express as a linear combination ?$

A.

Sieve of Eratosthenes

B.

Extended Euclidean Algorithm

C.

Fermat's Factorization

D.

Square and Multiply

41 $If, what is the last digit of ?$

A.

3

B.

7

C.

10

D.

Unknown

42 $A number is divisible by 3 if:$

A.

The last digit is divisible by 3

B.

The sum of its digits is divisible by 3

C.

It is odd

D.

The sum of its digits is divisible by 9

43 $Solve for : .$

A.

2

B.

7

C.

9

D.

4

44 $In cryptography, the set of integers represents:$

A.

The bit length of the key

B.

The ASCII codes

C.

The English alphabet letter values

D.

The prime factors of 26

45 $What is the result of ?$

A.

12

B.

7

C.

2

D.

1

46 $Which statement best describes a prime number ?$

A.

has exactly three divisors

B.

has exactly two distinct positive divisors: 1 and

C.

is any odd number

D.

is divisible by 2

47 $Using the Caesar cipher with (ROT13) on the letter 'A' results in:$

A.

M

B.

N

C.

Z

D.

O

48 $If, then equals:$

A.

B.

$1$

C.

D.

49 $Which of these equations is a Linear Congruence Equation?$

A.

B.

C.

D.

50 $Given . In CRT, we calculate . We need the inverse of modulo . Solve .$

A.

1

B.

2

C.

3

D.

4

51 $What is the smallest prime number?$

A.

B.

1

C.

2

D.

3

Unit 6 - Practice Quiz