Find the number of divisiors of 144 excluding itself.

Options

A. 13
B. 14
C. 15
D. 16

Answer

To find the number of divisors of 144 excluding itself, we can follow these steps:

### Step 1: Prime Factorization of 144
First, we need to find the prime factorization of 144.

144 can be divided as follows:
- 144 ÷ 2 = 72
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 144 is:
\[ 144 = 2^4 \times 3^2 \]

### Step 2: Use the Divisor Formula
The formula to find the number of divisors (d) from the prime factorization \( p_1^{e_1} \times p_2^{e_2} \times ... \times p_n^{e_n} \) is:
\[ d = (e_1 + 1)(e_2 + 1)...(e_n + 1) \]

For 144, we have:
- For \( 2^4 \), \( e_1 = 4 \)
- For \( 3^2 \), \( e_2 = 2 \)

Using the formula:
\[ d(144) = (4 + 1)(2 + 1) = 5 \times 3 = 15 \]

### Step 3: Exclude 144 Itself
Since we need the number of divisors excluding 144 itself, we subtract 1 from the total number of divisors:
\[ \text{Divisors excluding 144} = 15 - 1 = 14 \]

### Final Answer
The number of divisors of 144 excluding itself is **14**.

---