Finding the largest divider of big number without fractions

Question

Let's say I have a list of random numbers, all very big;

$$A: 1304810398401344441324$$

$$B: 5641893748137481374813764$$

$$C: 33322299992418948194899999$$

etc..

I want to be able to divide each and every number with another number, let's call it a divider, which will be unique for each number, however, I would like to find the biggest "divider" which will divide a number without leaving fractions.

How can I go about doing this in an efficient manner without trying every possible combination of random numbers???

to make it less complicated I'm making the "divider" at least $\frac 13$ as long as the number that's being divided by it.

Is this possible at all?

Thanks.

Answer 1

A thorough answer to your question is impossible on this site. Whether one can factor large numbers efficiently is an important open question. If you could do it much of the security on the internet would be at risk.

Your examples aren't particularly large for fast computers using the most efficient algorithms known. Wolfram alpha does the first one pretty much instantaneously, yielding $$ 1304810398401344441324 = 2^2×47×6940480842560342773 $$ https://www.wolframalpha.com/input/?i=factor+1304810398401344441324

You can start reading at

https://en.wikipedia.org/wiki/Integer_factorization

Answer 2

Just divide each $n$ by $n$ itself (or $n/2)$. That trivially satisfies all your stated requirements.