To find the number of positive factors of 132, we will follow these steps:
### Step 1: Prime Factorization of 132
First, we need to find the prime factorization of 132. We can do this by dividing 132 by the smallest prime numbers until we reach 1.
- Divide 132 by 2 (the smallest prime number):
\[
132 \div 2 = 66
\]
- Divide 66 by 2:
\[
66 \div 2 = 33
\]
- Now, divide 33 by 3 (the next smallest prime number):
\[
33 \div 3 = 11
\]
- Finally, 11 is a prime number, so we stop here.
Thus, the prime factorization of 132 is:
\[
132 = 2^2 \times 3^1 \times 11^1
\]
### Step 2: Use the Formula to Find the Number of Factors
To find the total number of positive factors of a number, we use the formula based on its prime factorization. If a number \( n \) can be expressed as:
\[
n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \ldots
\]
where \( p_1, p_2, p_3, \ldots \) are prime factors and \( e_1, e_2, e_3, \ldots \) are their respective powers, then the number of positive factors \( d(n) \) is given by:
\[
d(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots
\]
### Step 3: Apply the Formula
From the prime factorization \( 132 = 2^2 \times 3^1 \times 11^1 \):
- For \( 2^2 \), the exponent \( e_1 = 2 \), so \( e_1 + 1 = 2 + 1 = 3 \).
- For \( 3^1 \), the exponent \( e_2 = 1 \), so \( e_2 + 1 = 1 + 1 = 2 \).
- For \( 11^1 \), the exponent \( e_3 = 1 \), so \( e_3 + 1 = 1 + 1 = 2 \).
Now, we can calculate the total number of positive factors:
\[
d(132) = (2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2
\]
### Step 4: Calculate the Result
Now we multiply these results together:
\[
d(132) = 3 \times 2 \times 2 = 12
\]
Thus, the number of positive factors of 132 is **12**.
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