How do you prime factorize the numbers 8678670 and 15485863?
clearly, 10 (and thus 2 and 5) divide 8678670
8678670 = 2*5*867867
Now it is clear that
867867 = 867 * 1001
Since the digits of 867 sum to 21, it is divisible by 3
867 = 3*289
Now you just start testing primes, up to √289 = 17
Aha! now we are done with 867 = 17^2
As for 1001, start testing primes, up to √1001 ≈ 31
3,5,7 clearly do not divide 1001
11*91 = 1001
91 = 7*13
So 1001 =11*7*13
And so we know that
8678670 = (2*5)(3*17^2)(7*11*13) = 2•3•5•7•11•13•17^2
Do any others in the same way. Use casting out 3's and casting out 9's, repeated groups, and prime testing up to √N. Why stop there? Because if one factor is greater than √N, the other will be less than √N, so you will have already found it.