The prime factors of $\mathrm{13,195}$ are $5,7,13,$ and $29$.

What is the largest prime factor of $\mathrm{600,851,475,143}$?

2×3×5×7

The challenge is less about raw speed than about keeping the remaining number small as soon as a factor is known.

Hints

Open one at a time

Remove each small factor completely before continuing.

When the loop ends, the remaining number may itself be prime.

Solution

Best opened after a real attempt

Strip each factor completely

Whenever $d$ divides the remaining value, divide by $d$ until it no longer does. We only need to test while $d^2\le n$; if a value greater than 1 remains afterward, that remainder is prime.

function largestPrimeFactor(value: number): number {
  let n = value
  let largest = 1

  for (let divisor = 2; divisor * divisor <= n; divisor += divisor === 2 ? 1 : 2) {
    while (n % divisor === 0) {
      largest = divisor
      n /= divisor
    }
  }

  return n > 1 ? n : largest
}

For the given number, the result is

$$\boxed{{\displaystyle 6857}}.$$

The changing loop bound matters: every division reduces $n$, so later trial divisors do not need to approach the square root of the original input.