Number Theory and Divisibility Quiz

What does the fundamental theorem of arithmetic state about positive integers greater than 1?

• They can be factored into a product of even numbers.

• They can be factored into a product of composite numbers.

• They can be factored into a product of prime numbers in a unique way.

• They can be factored into a product of odd numbers.

Which of the following is true about prime numbers?

• They are divisible by 1 and themselves only.

• They are always even.

• They have more than two divisors.

• They are always greater than 10.

What is the tau function used for in number theory?

• To find the product of all divisors of a number.

• To count the number of divisors of a number.

• To find the sum of all divisors of a number.

• To determine if a number is prime.

How is a perfect number defined?

• A number whose sum of proper divisors is less than the number.

• A number whose sum of proper divisors is equal to the number.

• A number that is divisible by 10.

• A number whose sum of proper divisors is greater than the number.

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

• 12

• 9

• 28

• 15

Which of the following is true about deficient numbers?

• They have a sum of proper divisors greater than the number.

• They have a sum of proper divisors less than the number.

• They have a sum of proper divisors equal to the number.

• They are always even.

What is a Mersenne prime?

• A prime number that is a multiple of 5.

• A prime number that is one more than a power of 2.

• A prime number that is one less than a power of 2.

• A prime number that is a multiple of 3.

What is the relationship between even perfect numbers and Mersenne primes?

• Every even perfect number corresponds to a Mersenne prime.

• Every even perfect number is a sum of Mersenne primes.

• Every even perfect number is a multiple of a Mersenne prime.

• Every even perfect number is a difference of Mersenne primes.

What is still unknown about perfect numbers?

• Whether there are any odd perfect numbers.

• Whether perfect numbers can be negative.

• Whether perfect numbers can be fractions.

• Whether perfect numbers can be irrational.

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

• They alternated between 2 and 4.

• They alternated between 5 and 7.

• They alternated between 6 and 8.

• They alternated between 3 and 9.