Find the total no of factors of 60 ?

Options

A. A) 10
B. B) 13
C. C) 16
D. D) 12

Answer

To find the total number of factors of 60, we will follow these steps:

### Step 1: Prime Factorization of 60
First, we need to find the prime factorization of the number 60.

- Start by dividing 60 by the smallest prime number, which is 2:
- \( 60 \div 2 = 30 \)
- Continue dividing by 2:
- \( 30 \div 2 = 15 \)
- Now, 15 is not divisible by 2, so we move to the next prime number, which is 3:
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number itself:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 60 is:
\[ 60 = 2^2 \times 3^1 \times 5^1 \]

### Step 2: Use the Formula to Find the Total Number of Factors
To find the total number of factors, we use the formula based on the prime factorization. The formula states that if a number \( n \) can be expressed as:
\[ n = p_1^{e_1} \times p_2^{e_2} \times p_3^{e_3} \]
where \( p_1, p_2, p_3 \) are prime factors and \( e_1, e_2, e_3 \) are their respective powers, then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)(e_3 + 1) \]

For 60, we have:
- \( e_1 = 2 \) (for the prime factor 2)
- \( e_2 = 1 \) (for the prime factor 3)
- \( e_3 = 1 \) (for the prime factor 5)

### Step 3: Calculate the Total Number of Factors
Now we apply the formula:
\[ T(60) = (2 + 1)(1 + 1)(1 + 1) \]
\[ T(60) = 3 \times 2 \times 2 \]

Now, we calculate:
- \( 3 \times 2 = 6 \)
- \( 6 \times 2 = 12 \)

Thus, the total number of factors of 60 is **12**.

### Final Answer
The total number of factors of 60 is **12**.

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