Question

What is the sum of the factors of 27000?

• A. 11700
• B. 23400
• C. 46800
• D. 70200
• E. 93600

Answer 1

Answer: (E)

Step 1: Understand the problem.
We are asked to find the sum of the factors of 27000.

Step 2: Prime factorization of 27000.
To find the sum of the factors, we first need to perform the prime factorization of 27000.
We start by dividing 27000 by prime numbers:
\[ 27000 \div 2 = 13500 \quad \text{(divide by 2)} \] \[ 13500 \div 2 = 6750 \quad \text{(divide by 2)} \] \[ 6750 \div 2 = 3375 \quad \text{(divide by 2)} \] \[ 3375 \div 3 = 1125 \quad \text{(divide by 3)} \] \[ 1125 \div 3 = 375 \quad \text{(divide by 3)} \] \[ 375 \div 3 = 125 \quad \text{(divide by 3)} \] \[ 125 \div 5 = 25 \quad \text{(divide by 5)} \] \[ 25 \div 5 = 5 \quad \text{(divide by 5)} \] \[ 5 \div 5 = 1 \quad \text{(divide by 5)} \] Therefore, the prime factorization of 27000 is: \[ 27000 = 2^3 \times 3^3 \times 5^3 \]

Step 3: Use the formula for the sum of factors.
The formula for the sum of factors of a number \( n = p_1^{e_1} \times p_2^{e_2} \times \dots \times p_k^{e_k} \) is: \[ \text{Sum of factors of } n = (1 + p_1 + p_1^2 + \dots + p_1^{e_1})(1 + p_2 + p_2^2 + \dots + p_2^{e_2}) \dots (1 + p_k + p_k^2 + \dots + p_k^{e_k}) \] For \( 27000 = 2^3 \times 3^3 \times 5^3 \), the sum of the factors is: \[ \text{Sum of factors} = (1 + 2 + 2^2 + 2^3)(1 + 3 + 3^2 + 3^3)(1 + 5 + 5^2 + 5^3) \] Now, calculate each factor: \[ 1 + 2 + 2^2 + 2^3 = 1 + 2 + 4 + 8 = 15 \] \[ 1 + 3 + 3^2 + 3^3 = 1 + 3 + 9 + 27 = 40 \] \[ 1 + 5 + 5^2 + 5^3 = 1 + 5 + 25 + 125 = 156 \] Therefore, the sum of the factors is: \[ \text{Sum of factors} = 15 \times 40 \times 156 \] First, calculate \( 15 \times 40 = 600 \), and then: \[ 600 \times 156 = 93600 \]

Step 4: Conclusion.
The sum of the factors of 27000 is 93600.

Final Answer:
The correct answer is (E): 93600.