Sum of all odd proper divisors of 360 is:

Options

A. 78
B. 77
C. 75
D. 81

Answer

To find the sum of all odd proper divisors of 360, we can follow these steps:

### Step 1: Prime Factorization of 360
First, we need to factor 360 into its prime factors.

- 360 can be divided by 2:
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45 (now we cannot divide by 2 anymore)

Next, we divide by 3:
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5 (now we cannot divide by 3 anymore)

Finally, we divide by 5:
- 5 ÷ 5 = 1

So, the prime factorization of 360 is:
\[ 360 = 2^3 \times 3^2 \times 5^1 \]

### Step 2: Identify Odd Divisors
To find the odd divisors, we only consider the odd prime factors, which are 3 and 5. The even factor (2) will not contribute to odd divisors.

From the prime factorization:
- The odd part is \( 3^2 \times 5^1 \).

### Step 3: Calculate the Sum of Odd Divisors
To find the sum of the odd divisors, we can use the formula for the sum of divisors based on the prime factorization:
\[ \text{Sum of divisors} = (p_1^{k_1 + 1} - 1) / (p_1 - 1) \times (p_2^{k_2 + 1} - 1) / (p_2 - 1) \]

For \( 3^2 \) and \( 5^1 \):
- For \( 3^2 \):
\[ \text{Sum} = \frac{3^{2 + 1} - 1}{3 - 1} = \frac{3^3 - 1}{2} = \frac{27 - 1}{2} = \frac{26}{2} = 13 \]

- For \( 5^1 \):
\[ \text{Sum} = \frac{5^{1 + 1} - 1}{5 - 1} = \frac{5^2 - 1}{4} = \frac{25 - 1}{4} = \frac{24}{4} = 6 \]

### Step 4: Calculate Total Sum of Odd Divisors
Now we multiply the sums of the odd prime factors:
\[ \text{Total Sum of Odd Divisors} = 13 \times 6 = 78 \]

### Step 5: Exclude the Number Itself
Since we need the sum of proper divisors, we need to exclude 360 itself from the sum. However, since 360 is even, it does not affect the sum of odd proper divisors.

### Final Answer
Thus, the sum of all odd proper divisors of 360 is:
\[ \boxed{78} \]