Precisely List All Factors of 350: A Step-by-Step Guide

To solve this problem, we'll follow these steps:

• Step 1: Perform the prime factorization of $350$.
• Step 2: Identify all possible combinations of the factors.
• Step 3: Verify each step to ensure all factors are included.

Now, let's work through each step:
Step 1: Perform the prime factorization of $350$
The number $350$ is even, and hence divisible by $2$:
$350 \div 2 = 175$
Next, $175$ ends in $5$, indicating it is divisible by $5$:
$175 \div 5 = 35$
Next, $35$ is also divisible by $5$ (it ends in $5$):
$35 \div 5 = 7$
Finally, $7$ is a prime number.
Thus, the prime factorization of $350$ is $2 \times 5^2 \times 7$.

Step 2: Identify all possible combinations of factors.
Using the prime factors, combine them to list all factors of $350$:
- $1$ (trivial factor)
- $2$
- $5$
- $7$
- $2 \times 5 = 10$
- $2 \times 7 = 14$
- $5 \times 5 = 25$
- $5 \times 7 = 35$
- $2 \times 5^2 = 50$
- $2 \times 5 \times 7 = 70$
- $5^2 \times 7 = 175$
- $350$ (the number itself)

Step 3: Verify each step.
Verify each combination using basic multiplication to ensure accuracy.

Therefore, all the factors of $350$ are:
$1, 2, 5, 7, 10, 14, 25, 35, 50, 70, 175, 350$.