Deck 4: Number Theory and Cryptography

Question 1

Prove or disprove: For all integers

Answer 1

False: a = 1, b = −1.

Question 2

Find the prime factorization of 1,024.

Answer 2

Question 3

Prove or disprove: For all integers

Answer 3

False: a = b = c = 1.

Question 4

What does a 60-second stop watch read 54 seconds before it reads 19 seconds?

Question 5

Prove or disprove: For all integers

Question 6

What does a 60-second stop watch read 82 seconds after it reads 27 seconds?

Question 7

Find the prime factorization of 510,510.

Question 8

Find the prime factorization of 111,111.

Question 9

Prove or disprove: For all integers a, b, c, d, if a|b and c|d, then

Question 10

suppose that a and b are integers, such that

Question 11

Find the prime factorization of 8,827.

Question 12

suppose that a and b are integers, such that

Question 13

Prove or disprove: For all integers

Question 14

Prove or disprove: For all integers

Question 15

suppose that a and b are integers, such that

Question 16

Prove or disprove: For all integers

Question 17

Prove or disprove: For all integers

Question 18

suppose that a and b are integers, such that

Question 19

Find the prime factorization of 45,617.

Question 20

Find the prime factorization of 1,025.

Question 21

Prove or disprove: A positive integer congruent to 1 modulo 4 cannot have a prime factor congruent to 3
modulo 4.

Question 22

Find 289 mod 17.

Question 23

Applying the division algorithm

Question 24

Suppose that the lcm of two numbers is 400 and their gcd is 10. If one of the numbers is 50, find the other
number.

Question 25

Prove or disprove:

Question 26

Find −88 mod 13.

Question 27

Find

Question 28

Prove or disprove: The sum of two irrational numbers is irrational.

Question 29

Find Imc(20!, 12!) by directly finding the smallest positive multiple of both numbers

Question 30

List all positive integers less than 30 that are relatively prime to 20.

Question 31

Find gcd(2⁸⁹, 2³⁴⁶) by directly finding the largest divisor of both numbers.

Question 32

Find gcd(20!, 12!) by directly finding the largest divisor of both numbers.

Question 33

Find 18 mod 7.

Question 34

Find the hexadecimal expansion of

Question 35

Prove or disprove: If p and q are primes

Question 36

Prove or disprove: The sum of two primes is a prime.

Question 37

Prove or disprove: There exist two consecutive primes, each greater than 2.

Question 38

Find Icm(2⁸⁹, 2³⁴⁶) by directly finding the smallest positive multiple of both numbers.

Question 39

Find

Question 40

Prove or disprove:

Question 41

Find

Question 42

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 43

find each of these values
(123 mod 19 + 342 mod 19) mod 19.

Question 44

Find four integers

Question 45

Show that if a, b, k and m are integers such that

Question 46

Find three integers

Question 47

Find integers

Question 48

Find

Question 49

Find

Question 50

find each of these values

Question 51

Find the smallest

Question 52

find each of these values
(123 mod 19 · 342 mod 19) mod 19.

Question 53

Find an integer

Question 54

Find the integer a such that

Question 55

find each of these values

Question 56

Find

Question 57

Prove or disprove:

Question 58

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 59

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 60

Find

Question 61

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 62

Convert (2AC)16 to base 10.

Question 63

Explain in words the difference between a|b and

Question 64

Convert (1 1101)2 to base 16.

Question 65

Prove or disprove: if p and q are prime numbers, then pq + 1 is prime.

Question 66

Convert (271)₈ to base 2 .

Question 67

Convert (101011)₂ to base 8 .

Question 68

Convert (BC1)16 to base 2.

Question 69

(a) Find two positive integers, each with exactly three positive integer factors greater than 1.
(b) Prove that there are an infinite number of positive integers, each with exactly three positive integer factors
greater than 1.

Question 70

Convert (1 1101)2 to base 10.

Question 71

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 72

Convert (204)₁₀ to base 2 .

Question 73

Convert (8091)10 to base 2.

Question 74

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 75

Convert (6253)₈ to base 2 .

Question 76

Convert (100 1100 0011)₂ to base 16 .

Question 77

What sequence of pseudorandom numbers is generated using the pure multiplicative generator = mod 11 with seed ?

Question 78

Convert (10,000)10 to base 2.

Question 79

determine whether each of the following "theorems" is true or false. Assume that a, b, c,
d, and m are integers with m > 1.

Question 80

Eitherno suchfindinteger.an integer x such that x ≡ 2 (mod 6) and x ≡ 3 (mod 9) are both true, or else prove that there is