Find the number of factor of 1200

Options

A. 30
B. 24
C. 32
D. 36

Answer

To find the number of factors of 1200, we will follow these steps:

### Step 1: Prime Factorization of 1200
We start by finding the prime factorization of 1200.

1. Divide 1200 by 2:
\[ 1200 \div 2 = 600 \]
2. Divide 600 by 2:
\[ 600 \div 2 = 300 \]
3. Divide 300 by 2:
\[ 300 \div 2 = 150 \]
4. Divide 150 by 2:
\[ 150 \div 2 = 75 \]
5. Now, 75 is not divisible by 2, so we divide by 3:
\[ 75 \div 3 = 25 \]
6. Finally, divide 25 by 5:
\[ 25 \div 5 = 5 \]
7. And divide 5 by 5:
\[ 5 \div 5 = 1 \]

So, the prime factorization of 1200 is:
\[ 1200 = 2^4 \times 3^1 \times 5^2 \]

### Step 2: Using the Formula to Find the Number of Factors
To find the total number of factors of a number from its 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 powers of the prime factors.

From our factorization:
- The power of 2 is 4.
- The power of 3 is 1.
- The power of 5 is 2.

Now we apply the formula:
\[ \text{Number of factors} = (4 + 1)(1 + 1)(2 + 1) \]
Calculating each term:
- \( 4 + 1 = 5 \)
- \( 1 + 1 = 2 \)
- \( 2 + 1 = 3 \)

Now, multiply these together:
\[ 5 \times 2 \times 3 = 30 \]

### Final Answer
The number of factors of 1200 is **30**.

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