Find the number of factors of factors of the following :
1458

Answer

To find the number of factors of the number 1458, we will follow these steps:

### Step 1: Find the prime factorization of 1458
We start by dividing 1458 by the smallest prime number, which is 2.

- \( 1458 \div 2 = 729 \) (2 is a factor)
- Now, we will factor 729. Since 729 is odd, we try the next smallest prime, which is 3.
- \( 729 \div 3 = 243 \)
- \( 243 \div 3 = 81 \)
- \( 81 \div 3 = 27 \)
- \( 27 \div 3 = 9 \)
- \( 9 \div 3 = 3 \)
- \( 3 \div 3 = 1 \)

So, the prime factorization of 1458 is:
\[ 1458 = 2^1 \times 3^6 \]

### Step 2: Identify the powers of the prime factors
From the prime factorization, we have:
- The power of 2 is 1.
- The power of 3 is 6.

### Step 3: Use the formula to find the number of factors
The formula to find the number of factors of a number based on its prime factorization \( p_1^{e_1} \times p_2^{e_2} \) is:
\[ (e_1 + 1)(e_2 + 1) \]

In our case:
- \( e_1 = 1 \) (for the prime factor 2)
- \( e_2 = 6 \) (for the prime factor 3)

Now we can apply the formula:
\[ (1 + 1)(6 + 1) = 2 \times 7 = 14 \]

### Conclusion
Thus, the number of factors of 1458 is **14**.

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