Understanding Divisors and Prime Factorization

The video tutorial explains how to calculate the sum of all positive divisors of 27,000. It begins with the prime factorization of 27,000, breaking it down into 2^3, 3^3, and 5^3. The tutorial then explores combinations of divisors without including 5s, calculates their sums, and finally includes 5s to find the total sum of all divisors, resulting in 93,600.

What is the first step in solving the problem of finding the sum of all positive divisors of 27,000?

• Find the prime factorization of 27,000

• Divide 27,000 by 2

• List all numbers up to 27,000

• Calculate the square root of 27,000

What is the prime factorization of 27,000?

• 2^3 * 3^3 * 5^3

• 2^3 * 3^3 * 5^2

• 2^3 * 3^2 * 5^3

• 2^2 * 3^3 * 5^3

How many 2's can be in the divisors of 27,000?

• Up to four 2's

• Up to two 2's

• Up to three 2's

• Up to five 2's

What is the sum of divisors of 27,000 that do not include the factor of 5?

• 700

• 600

• 500

• 400

What happens to the sum of divisors when one factor of 5 is included?

• It is multiplied by 10

• It is multiplied by 5

• It remains the same

• It is divided by 5

What is the sum of divisors when two factors of 5 are included?

• 3,000

• 6,000

• 15,000

• 30,000

What is the sum of divisors when three factors of 5 are included?

• 45,000

• 60,000

• 75,000

• 90,000

What is the total sum of all positive divisors of 27,000?

• 113,600

• 103,600

• 93,600

• 83,600

How do you calculate the sum of all divisors including factors of 5?

• Divide the sum without 5's by 5

• Multiply the sum without 5's by 5

• Add 5 to the sum without 5's

• Subtract 5 from the sum without 5's

What is the significance of prime factorization in finding divisors?

• It helps in listing all numbers

• It is not significant

• It is used to find the square root

• It simplifies the calculation of divisors