Understanding Highly Composite Numbers

The video explores the concept of highly composite numbers, starting with Plato's preference for the number 5040 due to its numerous divisors. It introduces the idea of highly composite numbers, which have more divisors than any smaller number, and discusses the contributions of mathematician Ramanujan. The Fundamental Theorem of Arithmetic is explained, highlighting the role of prime numbers as building blocks. The video demonstrates how to calculate divisors and outlines properties of highly composite numbers, such as consecutive primes and decreasing powers. It concludes with a promotional segment for Audible.

Why did Plato consider 5040 to be an ideal number for organizing a city?

• It is a perfect square.

• It is a prime number.

• It is the smallest number divisible by 10.

• It has a large number of divisors.

What is a highly composite number?

• A number that is a power of 2.

• A number that is only divisible by 1 and itself.

• A number that is a perfect square.

• A number with more divisors than any smaller number.

Which of the following is NOT a highly composite number?

• 36

• 12

• 24

• 50

What is the significance of the number 60 in the context of highly composite numbers?

• It is a prime number.

• It is a perfect cube.

• It is the largest highly composite number.

• It is used in time measurement due to its divisibility.

What does the Fundamental Theorem of Arithmetic state?

• Every number is a multiple of 10.

• Every number is divisible by 1.

• Every positive whole number can be expressed as a product of prime numbers.

• Every number is a sum of two squares.

How many divisors does the number 30 have?

• 10

• 4

• 6

• 8

What is the prime factorization of 5040?

• 2^3 × 3^3 × 5 × 7

• 2^4 × 3^2 × 5 × 7

• 2^4 × 3 × 5^2 × 7

• 2^5 × 3 × 5 × 7

According to Ramanujan, what is a necessary condition for a number to be highly composite?

• It must have consecutive prime factors.

• It must be a perfect square.

• It must end with a prime factor raised to the power of three.

• It must be a prime number.

Which of the following numbers is an exception to Ramanujan's rule about highly composite numbers ending with a prime power of one?

• 60

• 24

• 12

• 4

What is the relationship between highly composite numbers and antiprime numbers?

• They are the same concept.

• Antiprime numbers have fewer divisors.

• Antiprime numbers are a humorous term for highly composite numbers.

• Highly composite numbers are a subset of antiprime numbers.