factorization of 71256?

Use a factor tree and divisibility rules:
The factors's digits add up to 7+1+2+5+6 = 21 which is divisible by 3.
It is even, so it is divisible by 2.
If it is divisible by 2 and by 3 like this, then it is divisible by 6.
Start a factor tree and continue it until you get to prime numbers.
..71256
... /\
6 ......11876
/\ ........./\
2 3 .....2 . 5938
..................... /\
............... 2 2969
So you have 2 * 2 * 2 * 3 * 2969 when you run out of factor rules for 1 - 10 and have to try to guess and check other numbers like 11, 13, etc.
I tried a bunch of prime numbers that don't have factors themselves up to 57 and they didn't work so you might be done here.

Do you mean prime factorization?
Because all it involves is breaking it down into prime numbers, using various hints.
In the case of 71256, it is an even number, so it is divisible by 2.

71256
2 x 35628
2 x 2 x 17814
2 x 2 x 2 x 8907

Since the digits add up to 24, which is divisible by 3, then 8907 is divisible by 3.

2 x 2 x 2 x 3 x 2969

And 2969 is prime, so we're done.

71256 = (2)(2)(2)(3)(2969)
2969 is prime

2^3 × 3 × 2969

71256 = 2 x 2 x 2 x 3 x 2969
or
71256 = 2^3 x 3 x 2969

71256 = (2)(2)(2)(3)(2969)

I think you are going about this wrong. Start with 71256 and find what integeres divides into it.

71256 is even so 2 divides it exactly
71256/2 = 35628
so (2) is a factor now repeat with 35628 until you find them all.

Prime factorization:
2^3*3*2969

All the divisors:
{ 1 2 3 4 6 8 12 24 2969 5938 8907 11876 17814 23752 35628 71256 }