The number of factors of 84 is

Options

A. 10
B. 18
C. 14
D. 12

Answer

To find the number of factors of the number 84, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of 84.

- 84 can be divided by 2 (the smallest prime number):
\[ 84 \div 2 = 42 \]
- Next, we can divide 42 by 2 again:
\[ 42 \div 2 = 21 \]
- Now, we divide 21 by 3 (the next smallest prime number):
\[ 21 \div 3 = 7 \]
- Finally, 7 is a prime number itself.

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

### Step 2: Use the Formula for Finding Number of Factors
To find the number of factors from the prime factorization, we use the formula:
\[ \text{Number of Factors} = (e_1 + 1)(e_2 + 1)(e_3 + 1) \ldots \]
where \(e_1, e_2, e_3, \ldots\) are the exponents in the prime factorization.

For 84:
- The exponent of 2 is 2, so \(e_1 = 2\).
- The exponent of 3 is 1, so \(e_2 = 1\).
- The exponent of 7 is 1, so \(e_3 = 1\).

Now, applying the formula:
\[ \text{Number of Factors} = (2 + 1)(1 + 1)(1 + 1) = 3 \times 2 \times 2 \]

### Step 3: Calculate the Result
Now, we calculate:
\[ 3 \times 2 = 6 \]
\[ 6 \times 2 = 12 \]

Thus, the number of factors of 84 is **12**.

### Summary
The number of factors of 84 is **12**.

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