Master Integer Factorization: A Key to Number Theory and Cryptography

Question 1

Find the prime factorization of 120.

Accepted Answer

2³ × 3 × 5

Answer

2² × 3² × 5

Answer

2 × 3 × 5²

Answer

2⁴ × 3 × 5

Question 2

Which number is NOT a factor of 36?

Accepted Answer

4

Answer

9

Answer

6

Answer

3

Question 3

Which of the following is a prime number?

Accepted Answer

11

Answer

9

Answer

15

Answer

12

Question 4

Describe the trial division method used in integer factorization.

Accepted Answer

Divide the integer consecutively by prime numbers until a divisor is found.

Answer

Multiply the integer consecutively by prime numbers until the product is the original integer.

Answer

Subtract prime numbers consecutively from the integer until the difference is the original integer.

Answer

Add prime numbers consecutively to the integer until the sum is the original integer.

Question 5

Select the application of integer factorization from the following:

Accepted Answer

Cryptography

Answer

Algebra

Answer

Geometry

Answer

Calculus

Question 6

Identify the limitation of integer factorization.

Accepted Answer

Factoring extremely large integers can be computationally demanding.

Answer

Integer factorization can solve all mathematical problems.

Answer

Integer factorization is always a straightforward process.

Answer

Integer factorization is applicable to all types of integers.

Question 7

Which method is commonly used for the factorization of large integers?

Accepted Answer

Trial division

Answer

Binomial theorem

Answer

Long division

Answer

Quadratic formula

Question 8

Identify the integer that is NOT a prime number:

Accepted Answer

12

Answer

7

Answer

11

Answer

5

Question 9

Determine the prime factorization of 36:

Accepted Answer

2^2 * 3^2

Answer

2 * 3 * 6

Answer

6 * 6

Answer

3 * 12

Question 10

Find the greatest common factor (GCF) of 12 and 18:

Accepted Answer

6

Answer

9

Answer

2

Answer

3

Question 11

Which application of integer factorization is employed in cryptography?

Accepted Answer

RSA encryption

Answer

AES encryption

Answer

DES encryption

Answer

SHA hashing

Question 12

Identify the perfect square among the following:

Accepted Answer

16

Answer

21

Answer

10

Answer

15

Question 13

Which of the following is a factor of 1,024?

Accepted Answer

32

Answer

63

Answer

42

Answer

25

Question 14

Identify the non-prime number among the following:

Accepted Answer

111

Answer

103

Answer

137

Question 15

Determine the greatest common factor (GCF) of 12 and 18.

Accepted Answer

6

Answer

9

Answer

3

Answer

12

Question 16

Which algorithm is suitable for efficient factorization of large integers?

Accepted Answer

Fermat's Factorization Method

Answer

Binary Search

Answer

Insertion Sort

Answer

Bubble Sort

Question 17

What's the primary application of integer factorization in cryptography?

Accepted Answer

Breaking encryption and discovering secret keys

Answer

Optimizing data storage

Answer

Solving complex equations

Answer

Generating random numbers

Question 18

Identify a potential challenge in integer factorization:

Accepted Answer

Computational complexity

Answer

Non-unique solutions

Answer

Limited applications

Answer

Lack of algorithms

Question 19

What's the use of Fermat's Little Theorem in integer factorization?

Accepted Answer

Discovering factors of composite numbers

Answer

Determining primality

Answer

Generating random numbers

Question 20

Which technique improves the efficiency of trial division?

Accepted Answer

Wheel factorization

Answer

Quicksort

Answer

Binary search

Answer

Bubble sort

Question 21

Which of the following numbers is composite?

Accepted Answer

15

Answer

11

Answer

7

Answer

13

Question 22

What is the primary application of integer factorization in cryptography?

Accepted Answer

Breaking codes and deciphering encrypted messages.

Answer

Generating random numbers

Answer

Solving mathematical puzzles

Answer

Compressing data

Question 23

Which of the following is NOT a factor of 360?

Accepted Answer

11

Answer

2

Answer

5

Answer

3

Question 24

If p and q are prime numbers and n = p * q, then the number of positive divisors of n is:

Accepted Answer

(p + 1) * (q + 1)

Answer

p^2 * q^2

Answer

p + q

Answer

p * q

Question 25

State the fundamental theorem of arithmetic.

Accepted Answer

Every positive integer greater than 1 can be uniquely factorized as a product of prime numbers.

Answer

Every even integer can be written as the sum of two primes.

Answer

The product of two prime numbers is always prime.

Question 26

Determine whether the integer 101 is prime or composite.

Accepted Answer

Prime

Answer

Composite

Answer

Neither prime nor composite

Answer

Semi-prime

Question 27

Which of the following is a limitation of using trial division for integer factorization?

Accepted Answer

Inefficient for large integers

Answer

Requires knowledge of all prime numbers

Answer

Always provides the prime factorization

Answer

Difficult to implement

Question 28

Integer factorization is a fundamental technique in which area of mathematics?

Accepted Answer

Number theory and cryptography

Answer

Algebra and geometry

Answer

Calculus and trigonometry

Answer

Statistics and probability

Question 29

Find the greatest common factor (GCF) of 24 and 36.

Accepted Answer

12

Answer

18

Answer

8

Answer

6

Question 30

Which of the following is a prime number?

Accepted Answer

17

Answer

12

Answer

24

Answer

9

Question 31

Find the prime factorization of 72.

Accepted Answer

2^3 * 3^2

Answer

3 * 4 * 6

Answer

2 * 2 * 3 * 6

Answer

2 * 3 * 12

Question 32

Which option represents the prime factorization of 120?

Accepted Answer

2^3 * 3 * 5

Answer

2^2 * 3 * 5 * 7

Answer

2 * 3^2 * 5

Answer

2 * 3 * 5 * 10

Question 33

Using prime factorization, find the greatest common divisor (GCD) of 48 and 72.

Accepted Answer

24

Answer

12

Answer

144

Answer

36

Question 34

Which of the following is the prime factorization of 120, expressed in index notation?

Accepted Answer

2³ × 3 × 5

Answer

2² × 3² × 5

Answer

2 × 3 × 5 × 10

Answer

2⁴ × 3 × 5

Question 35

What is the greatest common divisor (GCD) of 72 and 90, calculated using the Euclidean algorithm?

Accepted Answer

18

Answer

2

Answer

36

Answer

54

Question 36

Using trial division, determine the largest prime factor of 133.

Accepted Answer

11

Answer

7

Answer

19

Answer

3

Question 37

If a number is divisible by both 3 and 4, is it necessarily divisible by 12?

Accepted Answer

Yes

Answer

Only if the number is odd.

Answer

No, it could be divisible by 6 but not 12.

Answer

Only if the number is even.

Question 38

Which of the following numbers is a perfect square and a multiple of 5?

Accepted Answer

100

Answer

49

Answer

144

Answer

81

Question 39

Determine the least common multiple (LCM) of 12, 15, and 18.

Accepted Answer

180

Answer

270

Answer

90

Answer

30

Question 40

If a number is divisible by 9, is it necessarily divisible by 3?

Accepted Answer

Yes

Answer

Only if the number is prime.

Answer

Only if the number is odd.

Answer

No, the number could be divisible by 6 but not 3.

Question 41

Find the product of the prime factors of 45.

Accepted Answer

225

Answer

90

Answer

135

Answer

405

Question 42

Identify all the factors of 180.

Accepted Answer

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

Answer

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 20, 30, 60, 90, 180

Answer

1, 2, 3, 4, 6, 9, 10, 15, 20, 30, 45, 90, 180

Answer

1, 2, 3, 5, 6, 10, 12, 15, 18, 30, 36, 90, 180

Question 43

Explain why the number 1 is not considered a prime number.

Accepted Answer

1 is divisible by itself and 1, violating the definition of a prime number.

Answer

1 is an even number.

Answer

1 is not a factor of any other number.

Answer

1 is a large number.

Question 44

What is the prime factorization of 120?

Accepted Answer

2³ × 3 × 5

Answer

2 × 3² × 5

Answer

2² × 3 × 5

Answer

2 × 3 × 5²

Question 45

Which of the following is a factor of 144 but not a prime factor?

Accepted Answer

12

Answer

2

Answer

16

Answer

3

Question 46

What is the greatest common factor (GCD) of 36 and 60?

Accepted Answer

12

Answer

18

Answer

30

Answer

6

Question 47

Which of the following methods is NOT a common method for integer factorization?

Accepted Answer

Linear Programming

Answer

Trial Division

Answer

Fermat's Factorization Method

Answer

Pollard's Rho Method

Question 48

Using trial division, what is the smallest prime factor of 101?

Accepted Answer

101

Answer

2

Answer

7

Answer

5

Question 49

What is the least common multiple (LCM) of 12 and 18?

Accepted Answer

36

Answer

6

Answer

24

Answer

72

Question 50

Which of the following numbers is a perfect square?

Accepted Answer

144

Answer

169

Answer

121

Question 51

If a number is divisible by both 2 and 3, it must also be divisible by which of the following?

Accepted Answer

6

Answer

12

Answer

4

Answer

9

Question 52

Which of the following is a prime number?

Accepted Answer

17

Answer

21

Answer

25

Answer

9

Question 53

A student uses Fermat's Factorization Method to factor the number 143. What are the two factors they will find?

Accepted Answer

11 and 13

Answer

5 and 29

Answer

3 and 47

Answer

7 and 21