In how many ways can you resolve 846 in to two factors?

Options

A. 12
B. 6
C. 3
D. 8

Answer

To find the number of ways to resolve 846 into two factors, we can follow these steps:

### Step 1: Factorization of 846
First, we need to find the prime factorization of 846.

846 can be divided by 2 (since it is even):
\[ 846 \div 2 = 423 \]

Next, we factor 423. It is not divisible by 2, but it is divisible by 3:
\[ 423 \div 3 = 141 \]

Now, we factor 141. It is again divisible by 3:
\[ 141 \div 3 = 47 \]

Finally, 47 is a prime number.

Thus, the prime factorization of 846 is:
\[ 846 = 2^1 \times 3^2 \times 47^1 \]

### Step 2: Finding the Total Number of Factors
To find the total number of factors of a number, we use the formula:
If \( n = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k} \), then the total number of factors \( T(n) \) is given by:
\[ T(n) = (e_1 + 1)(e_2 + 1)...(e_k + 1) \]

For 846:
- The exponent of 2 is 1, so \( e_1 + 1 = 1 + 1 = 2 \)
- The exponent of 3 is 2, so \( e_2 + 1 = 2 + 1 = 3 \)
- The exponent of 47 is 1, so \( e_3 + 1 = 1 + 1 = 2 \)

Thus, the total number of factors is:
\[ T(846) = (1 + 1)(2 + 1)(1 + 1) = 2 \times 3 \times 2 = 12 \]

### Step 3: Resolving into Two Factors
Each factor can be paired with another factor to form the product 846. However, each pair (a, b) is the same as (b, a). Therefore, to find the number of unique pairs of factors, we divide the total number of factors by 2.

Thus, the number of ways to resolve 846 into two factors is:
\[ \text{Number of ways} = \frac{T(846)}{2} = \frac{12}{2} = 6 \]

### Conclusion
Therefore, 846 can be resolved into two factors in **6 ways**.

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