Number Theory: Master the Fundamentals of Number Properties

Question 1

Which of the following is a prime number?

Accepted Answer

13

Answer

6

Answer

24

Answer

35

Question 2

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

Accepted Answer

6

Answer

2

Answer

3

Answer

9

Question 3

Determine whether the number 36 is a composite number.

Accepted Answer

True

Answer

False

Question 4

Determine the greatest common factor (GCF) of the following numbers: 24, 36, and 48.

Accepted Answer

12

Answer

4

Answer

6

Question 5

Identify the divisibility rule for numbers ending in 5 and illustrate its application with an example.

Accepted Answer

A number is divisible by 5 if its last digit is 0 or 5.

Answer

A number is divisible by 5 if the sum of its digits is divisible by 5.

Answer

A number is divisible by 5 if the difference between its first and last digits is divisible by 5.

Question 6

Identify all the factors of the number 30:

Accepted Answer

1, 2, 3, 5, 6, 10, 15, 30

Answer

2, 3, 5

Answer

2, 5, 10

Answer

1, 2, 3, 6, 15, 30

Question 7

Which of the following numbers is a composite number?

Accepted Answer

14

Answer

7

Answer

11

Answer

19

Question 8

How does number theory play a role in the field of cryptography?

Accepted Answer

Encrypting and decrypting messages using prime factorization

Answer

Generating random numbers for simulations and games

Answer

Calculating the area of geometric shapes

Question 9

What is the least common multiple (LCM) of 4 and 6?

Accepted Answer

12

Answer

8

Answer

10

Answer

14

Question 10

Which of the following is a composite number?

Accepted Answer

9

Answer

7

Answer

11

Answer

13

Question 11

Which of the following is a necessary condition for two numbers to be relatively prime?

Accepted Answer

They have no common factors other than 1.

Answer

They are both even.

Answer

Their sum is prime.

Answer

Their product is prime.

Question 12

Which of the following numbers is NOT a prime number?

Accepted Answer

9

Answer

11

Answer

7

Answer

5

Question 13

Which of the following numbers is a composite number?

Accepted Answer

21

Answer

13

Answer

17

Answer

23

Question 14

What is the remainder when 25 is divided by 7?

Accepted Answer

4

Answer

3

Answer

5

Answer

6

Question 15

Which of the following statements about the number 1 is true?

Accepted Answer

It is neither prime nor composite

Answer

It is both a prime and a composite number

Answer

It is a prime number

Answer

It is a composite number

Question 16

Which of the following statements about prime numbers is true?

Accepted Answer

A prime number is a positive integer greater than 1 that has exactly two distinct factors: 1 and itself.

Answer

A prime number is a number that is not divisible by any other number.

Answer

A prime number is a number that can only be divided by itself and 2.

Question 17

Which of the following is a property of divisibility by 9?

Accepted Answer

The sum of its digits is divisible by 9.

Answer

The number of factors is divisible by 9.

Answer

The last digit is divisible by 9.

Question 18

Which of the following is NOT a prime number?

Accepted Answer

15

Answer

13

Answer

19

Answer

17

Question 19

Which of the following is a composite number?

Accepted Answer

9

Answer

7

Answer

11

Question 20

The prime factorization of 100 is:

Accepted Answer

2^2 * 5^2

Answer

2 * 3 * 5 * 7

Answer

5 * 10 * 2

Answer

10 * 10

Question 21

The number of divisors of 12 is:

Accepted Answer

6

Answer

9

Answer

3

Answer

12

Question 22

Which of the following is NOT a standard divisibility rule?

Accepted Answer

Remainder theorem

Answer

Divisibility by 2 (even number)

Answer

Divisibility by 3 (sum of digits divisible by 3)

Answer

Divisibility by 5 (ends in 0 or 5)

Question 23

Which of the following numbers is a prime number?

Accepted Answer

7

Answer

12

Answer

9

Answer

15

Question 24

What is the greatest common factor (GCF) of 12 and 18?

Accepted Answer

6

Answer

4

Answer

2

Answer

9

Question 25

What is the least common multiple (LCM) of 6 and 8?

Accepted Answer

24

Answer

12

Answer

16

Question 26

What is a common factor of both 15 and 21?

Accepted Answer

3

Answer

7

Answer

5

Answer

10

Question 27

What is the remainder when 25 is divided by 7?

Accepted Answer

4

Answer

3

Answer

1

Answer

5

Question 28

Prove that the sum of two even numbers is always even. Use the definition of even numbers in your proof.

Accepted Answer

Let 'a' and 'b' represent two even numbers. By definition, there exist integers 'p' and 'q' such that a = 2p and b = 2q. Therefore, a + b = 2p + 2q = 2(p + q). Since (p + q) is also an integer, the result 2(p + q) represents an even number.

Answer

The sum of two even numbers is always even because even numbers are divisible by 2.

Question 29

Which of the following sets of numbers is NOT closed under addition? (Adding any two numbers in the set must result in a number that is also in the set.)

Accepted Answer

The set of odd numbers

Answer

The set of even numbers

Answer

The set of real numbers

Answer

The set of rational numbers

Question 30

Which of the following is a necessary condition for two numbers to be relatively prime?

Accepted Answer

Their greatest common factor is 1.

Answer

Their sum is even.

Answer

Their product is a prime number.

Answer

Their difference is prime.

Question 31

List all the positive divisors of 12.

Accepted Answer

1, 2, 3, 4, 6, 12

Answer

1, 2, 3, 5, 6, 10, 12

Answer

1, 2, 3, 4, 5, 6, 12

Answer

1, 2, 3, 4, 6

Question 32

Which of the following numbers is a prime number?

Accepted Answer

17

Answer

21

Answer

9

Answer

15

Question 33

What is the greatest common factor (GCF) of 12 and 18?

Accepted Answer

6

Answer

2

Answer

4

Answer

8

Question 34

Which of the following numbers is divisible by 5?

Accepted Answer

25

Answer

13

Answer

17

Answer

19

Question 35

Calculate the least common multiple (LCM) of 6 and 8.

Accepted Answer

24

Answer

12

Answer

16

Answer

32

Question 36

Which of the following numbers is a composite number?

Accepted Answer

12

Answer

11

Answer

5

Answer

7

Question 37

What is the prime factorization of 36?

Accepted Answer

2^2 x 3^2

Answer

3 x 4 x 6

Answer

2 x 3 x 6

Question 38

What is the remainder when 19 is divided by 4?

Accepted Answer

3

Answer

2

Answer

0

Answer

1

Question 39

Which of the following numbers is a perfect square?

Accepted Answer

64

Answer

50

Answer

25

Answer

36

Question 40

What is the primary purpose of the Euclidean Algorithm?

Accepted Answer

Finding the GCF of two numbers

Answer

Determining divisibility

Answer

Solving quadratic equations

Answer

Finding the LCM of two numbers

Question 41

Which of the following is known as a fundamental theorem of arithmetic?

Accepted Answer

Every positive integer can be expressed as a unique product of primes.

Answer

The GCF and LCM of two numbers are always equal.

Answer

The sum of two even numbers is always even.

Question 42

Which of the following numbers is a prime number?

Accepted Answer

17

Answer

36

Answer

24

Answer

12

Question 43

What is the greatest common factor (GCD) of 18 and 24?

Accepted Answer

6

Answer

12

Answer

2

Answer

9

Question 44

Which of the following is a composite number?

Accepted Answer

20

Answer

2

Answer

7

Answer

3

Question 45

What is the least common multiple (LCM) of 6 and 8?

Accepted Answer

24

Answer

48

Answer

12

Answer

36

Question 46

If a number is divisible by both 2 and 3, it is also divisible by:

Accepted Answer

6

Answer

9

Answer

4

Answer

5

Question 47

What is the prime factorization of 36?

Accepted Answer

2² x 3²

Answer

2 x 3 x 6

Answer

4 x 9

Answer

3 x 12

Question 48

Which of the following is a perfect square?

Accepted Answer

81

Answer

25

Answer

40

Answer

30

Question 49

The sum of the first five prime numbers is:

Accepted Answer

28

Answer

35

Answer

17

Answer

10

Question 50

A number is divisible by 9 if the sum of its digits is divisible by:

Accepted Answer

9

Answer

12

Answer

3

Answer

6

Question 51

The greatest common factor (GCD) of two numbers is 1. What can you conclude about these numbers?

Accepted Answer

They are relatively prime.

Answer

They are both prime.

Answer

They are both odd.

Answer

They are both even.