Regular math vs competition math

TL;DR

Factors of a number can be written in the form of 2 to some power times 5 to some power, and the sum of divisors can be calculated using the fundamental theorem of arithmetic.

Transcript

Regular math compute the sum of all positive divisors of 186 1 times 186 is equal to 186 and we can  also say 2 times 93 and 3 times 62 and   lastly we can also have 6 times 31 and if we  sum them up we will get 284 and we are done competition math compute the sum of all positive integers whose positive divisors sum to 186 and this is not the same ... Read More

Key Insights

• 🍹 The sum of divisors of a number can be calculated using prime factorization and the fundamental theorem of arithmetic.
• ✊ Numbers can be written as a product of prime factors raised to specific powers.
• ✊ The arrangement of prime factors and their powers determines the sum of divisors for a given number.
• 🪜 Some numbers can have divisors that add up to a specific value, while others cannot.

Questions & Answers

Q: How can the sum of positive divisors of a number be computed efficiently?

The sum of divisors can be calculated by prime factorizing the number and using the fundamental theorem of arithmetic, which involves adding powers of the prime factors.

Q: Is it possible to find numbers whose divisors sum to a specific value?

Yes, by rearranging and identifying prime factors, it is possible to find numbers whose divisors add up to a desired value. Prime factors are used to determine the powers in the formulas.

Q: How can the sum of divisors of a number that gives a certain value be found?

By analyzing the prime factorization of the desired sum value, it is possible to identify the prime factors and their powers that will result in the desired sum when plugged into the formula.

Q: Can any number have a sum of divisors that equals a given value?

No, not all numbers can have divisors that sum to a specific value. The ability to find such numbers depends on the factorization and arrangement of prime factors.

Summary & Key Takeaways

• The video explains how to compute the sum of positive divisors of a number and how to find numbers whose divisors sum to a specific value.

• Observations are made using the example of the number 40 and its divisors.

• The fundamental theorem of arithmetic is introduced and used to derive a formula for the sum of divisors of a number.