Question 1

Write the number as the sum of two primes. There may be more than one way to do this.
-28

Accepted Answer

The sum 5 + 23 and 11 + 17 both equal 28, and all numbers involved are prime numbers.

Question 2

List all the abundant numbers between 20 and 30 (inclusive)

Accepted Answer

The answer of List all the abundant numbers between 20...

Question 3

Fermat proved that every odd prime number can be expressed as the difference of two
squares in one and only one way. Express each of the first 6 odd prime numbers as the
difference of two squares.

Accepted Answer

The answer of Fermat proved that every odd prime number...

Question 4

Write the number as the sum of two primes. There may be more than one way to do this.
-12

Accepted Answer

5 and 7 are both prime numbers, and their sum equals 12. The other options include numbers that are not prime: 10, 9, and 6.

Question 5

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair.
-2620 and 2924

Accepted Answer

The answer of For the following amicable pair, determine whether...

Question 6

Show that the number 132 is not perfect.

Accepted Answer

The answer of Show that the number 132 is not...

Question 7

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-32 and 39

Accepted Answer

Twin primes are pairs of primes that are exactly two units apart. Among the options, only 37 is a prime number, but 39 is not. Therefore, there are no twin primes between 32 and 39.

Question 8

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair.
-79,750 and 88,730

Accepted Answer

A happy number is a number which eventually reaches 1 when replaced by the sum of the square of each digit. Neither 79,750 nor 88,730 are happy numbers, as neither reaches 1 through this process. Therefore, the pair is not a happy amicable pair.

Question 9

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-15 and 24

Accepted Answer

Twin primes are pairs of primes that are two units apart. Between 15 and 24, the only twin primes are 17 and 19.

Question 10

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-34 and 49

Accepted Answer

Twin primes are pairs of prime numbers that have a difference of 2. Between 34 and 49, the pair 41 and 43 are both prime numbers and have a difference of 2, making them twin primes.

Question 11

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-4 and 16

Accepted Answer

Twin primes are pairs of primes that differ by 2. Between 4 and 16, the pairs of twin primes are (5, 7) and (11, 13).

Question 12

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair.
-32,685,250 and 34,538,270

Accepted Answer

Both members are happy because the sum of the proper divisors of 32,685,250 is 34,538,270, and the sum of the proper divisors of 34,538,270 is 32,685,250.

Question 13

Write the number as the sum of two primes. There may be more than one way to do this.
-18

Accepted Answer

18 can be written as the sum of two primes in two ways: 5 + 13 and 7 + 11. 15 is not a prime number, and 9 is not a prime number either.

Question 14

Write the number as the sum of two primes. There may be more than one way to do this.
-34

Accepted Answer

All pairs in option A are valid since each pair sums up to 34 and consists of prime numbers only.

Question 15

Solve the problem.
-Primorial primes are those of the form $p \# \pm 1$.
Apply this formula to the value $p = 11$ to obtain two numbers and state whether both, neither, or exactly one of these numbers is prime.

Accepted Answer

The answer of Solve the problem.
-Primorial primes are those of...

Question 16

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-80 and 100

Accepted Answer

Twin primes are pairs of primes that are two units apart (e.g., 11 and 13, 17 and 19). Among the options given, none are pairs of prime numbers. 89 is a prime number, but 91 is divisible by 7 and 13. 83 is prime, but 85 is divisible by 5. 91 and 93 are not prime; 91 is divisible by 7 and 13, and 93 is divisible by 3. Therefore, there are no twin primes between 80 and 100.

Question 17

Solve the problem.
-Factorial primes are those of the form n! ± 1 for natural numbers n. Determine the two numbers that are obtained when this formula is applied to the value n = 6, and state whether both, neither, or exactly one of them is prime.

Accepted Answer

The answer of Solve the problem.
-Factorial primes are those of...

Question 18

Determine whether the number is abundant or deficient.
-34

Accepted Answer

A number is considered deficient if the sum of its proper divisors (excluding itself) is less than the number itself. For 34, the proper divisors are 1, 2, 17, and their sum is 20, which is less than 34, making it deficient.

Question 19

Write the number as the sum of two primes. There may be more than one way to do this.
-44

Accepted Answer

Option B is correct because all pairs (3 + 41, 7 + 37, 13 + 31) consist of prime numbers that sum up to 44. Option A is incorrect because 22 is not a prime number. Option C includes 11 + 33, but 33 is not a prime number. Option D includes 5 + 39, but 39 is not a prime number.

Question 20

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes.
-99 and 107

Accepted Answer

Twin primes are pairs of primes that are two units apart. Among the options, only 101 and 103 are both prime and two units apart.

Quiz 5: Number Theory

Write the number as the sum of two primes. There may be more than one way to do this. -28

List all the abundant numbers between 20 and 30 (inclusive)

Fermat proved that every odd prime number can be expressed as the difference of two squares in one and only one way. Express each of the first 6 odd prime numbers as the difference of two squares.

Write the number as the sum of two primes. There may be more than one way to do this. -12

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -2620 and 2924

Show that the number 132 is not perfect.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -32 and 39

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -79,750 and 88,730

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -15 and 24

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -34 and 49

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -4 and 16

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -32,685,250 and 34,538,270

Write the number as the sum of two primes. There may be more than one way to do this. -18

Write the number as the sum of two primes. There may be more than one way to do this. -34

Solve the problem. -Primorial primes are those of the form $p \# \pm 1$ . Apply this formula to the value $p = 11$ to obtain two numbers and state whether both, neither, or exactly one of these numbers is prime.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -80 and 100

Solve the problem. -Factorial primes are those of the form n! ± 1 for natural numbers n. Determine the two numbers that are obtained when this formula is applied to the value n = 6, and state whether both, neither, or exactly one of them is prime.

Determine whether the number is abundant or deficient. -34

Write the number as the sum of two primes. There may be more than one way to do this. -44

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -99 and 107

Quiz 5: Number Theory

Write the number as the sum of two primes. There may be more than one way to do this. -28

List all the abundant numbers between 20 and 30 (inclusive)

Fermat proved that every odd prime number can be expressed as the difference of two squares in one and only one way. Express each of the first 6 odd prime numbers as the difference of two squares.

Write the number as the sum of two primes. There may be more than one way to do this. -12

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -2620 and 2924

Show that the number 132 is not perfect.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -32 and 39

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -79,750 and 88,730

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -15 and 24

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -34 and 49

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -4 and 16

For the following amicable pair, determine whether neither, one, or both of the members are happy, and whether the pair is a happy amicable pair. -32,685,250 and 34,538,270

Write the number as the sum of two primes. There may be more than one way to do this. -18

Write the number as the sum of two primes. There may be more than one way to do this. -34

Solve the problem. -Primorial primes are those of the form p#±1p \# \pm 1p#±1 . Apply this formula to the value p=11p = 11p=11 to obtain two numbers and state whether both, neither, or exactly one of these numbers is prime.

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -80 and 100

Solve the problem. -Factorial primes are those of the form n! ± 1 for natural numbers n. Determine the two numbers that are obtained when this formula is applied to the value n = 6, and state whether both, neither, or exactly one of them is prime.

Determine whether the number is abundant or deficient. -34

Write the number as the sum of two primes. There may be more than one way to do this. -44

Determine whether or not one or more pairs of twin primes exist between the pair of numbers given. If so, identify the twin primes. -99 and 107

Solve the problem. -Primorial primes are those of the form $p \# \pm 1$ . Apply this formula to the value $p = 11$ to obtain two numbers and state whether both, neither, or exactly one of these numbers is prime.