Sum of factors of 27000 | AIME | Math for fun and glory | Khan Academy | Summary and Q&A

TL;DR

The sum of all positive divisors of 27,000 is 93,600.

Key Insights

• #️⃣ The prime factorization of a number provides insights into the structure of its divisors.
• ❓ Divisors of 27,000 can have a combination of up to three 2's, three 3's, and three 5's.
• 🧑‍🏭 By organizing divisors based on their factors, the sum can be calculated efficiently.
• 🍹 Multiplying the sum of divisors without 5's by powers of 5 yields the sums for divisors with 1, 2, and 3 5's.

Transcript

Calculate the sum of all positive divisors of 27,000. The easiest thing that I can think of doing is first take the prime factorization of 27,000, and then that will help us kind of structure our thought of what all of the different divisors of 27,000 would have to look like. So 27,000 is the same thing as 27 times 1,000, which is the same thing as... Read More

Questions & Answers

Q: How can we determine the prime factorization of 27,000?

The prime factorization of 27,000 is 2^3 * 3^3 * 5^3, which means it can be written as the product of three 2's, three 3's, and three 5's.

Q: What is the significance of prime factorization in finding divisors?

Prime factorization helps us understand the structure of divisors. In the case of 27,000, any divisor must be a combination of powers of 2, 3, and 5.

Q: How can we calculate the sum of divisors without finding each individual divisor?

By organizing divisors based on the number of 2's and 3's they contain, we can quickly calculate the sum. Each row represents divisors with a specific number of 2's, while each column represents divisors with a specific number of 3's.

Q: How can we calculate the overall sum of divisors?

By summing the rows, we find that the sum for each row is 40, 80, 160, and 320. Considering the assumption of no 5's, the total sum of all divisors is calculated as 40 + 80 + 160 + 320 = 600.

Summary & Key Takeaways

• 27,000 can be prime factorized as 2^3 * 3^3 * 5^3.

• Any divisor of 27,000 can be made up of up to three 2's, up to three 3's, and up to three 5's.

• By considering all possible combinations, the sum of all positive divisors of 27,000 is found to be 93,600.