Largest Prime Factor - Insight

You do not need to find all prime factors and then pick the largest. Instead, you can divide out small factors as you find them.

Start with $i = 2$. While $i$ divides $n$, divide $n$ by $i$. Then increment $i$. After all small factors are removed, what remains is either $1$ or the largest prime factor.

Why does this work? If $n$ has a composite factor, its prime factors are smaller and will be divided out first. The largest factor left is always prime.