Number Theory and Divisibility Quiz

What does the fundamental theorem of arithmetic state about positive integers?

• They are always even numbers.

• They can be divided by 1 and themselves only.

• They cannot be factored into smaller numbers.

• They can be uniquely factored into prime numbers.

Which number is considered a unit and not a prime?

• 5

• 1

• 3

• 2

What is the sigma function used for in number theory?

• Summing all integers that divide a given integer.

• Counting the number of prime numbers.

• Finding the largest divisor of a number.

• Determining if a number is even or odd.

How is the tau function different from the sigma function?

• It counts the number of divisors, not their sum.

• It only applies to prime numbers.

• It finds the largest divisor of a number.

• It is used to determine if a number is perfect.

What is a perfect number?

• A number that is a power of two.

• A number equal to the sum of its proper divisors.

• A number with no divisors.

• A number that is only divisible by itself.

Which of the following is an example of a perfect number?

• 28

• 12

• 15

• 9

What pattern was observed in the last digits of early perfect numbers?

• They ended in zero.

• They were all prime numbers.

• They alternated between six and eight.

• They were all odd.

What is a Mersenne prime?

• A prime number that is one less than a power of two.

• A prime number that is a power of three.

• A composite number that is one less than a power of two.

• A number that is both prime and even.

What is the relationship between Mersenne primes and perfect numbers?

• Every Mersenne prime corresponds to a perfect number.

• Mersenne primes are always perfect numbers.

• Perfect numbers are always Mersenne primes.

• There is no relationship between them.

What is currently unknown about odd perfect numbers?

• Whether they exist.

• If they are all even.

• If they are greater than 100.

• Whether they are divisible by 3.