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

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December 30, 2010
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Sum of factors of 27000 | AIME | Math for fun and glory | Khan Academy

TL;DR

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

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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.

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