How would you go about finding prime factors of a number like $7999973$? I have trivial knowledge about divisor-searching algorithms.

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    $\begingroup$ This is the only context you have to solve the problem. Find a prime factor of 7999973 without a calculator. I got this problem from an old problem book, but there is no explanation or further context. $\endgroup$ – user523628 Jan 21 '18 at 23:52
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    $\begingroup$ Perhaps you didn't read the question carefully... or perhaps you didn't write the question carefully. Finding prime factors (implicitly: all prime factors) of the number, without a calculator, is going to be quite hard. Finding a prime factor (that is: only one asked for), as in your title, is quite easy in this case. $\endgroup$ – David Jan 21 '18 at 23:57
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    $\begingroup$ If ever I could understand what is special with this question and such votes $\endgroup$ – Guy Fsone Jan 22 '18 at 21:27
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    $\begingroup$ Those who don't understand this problem might understand it more if they assumed that it's a problem that is meant to be solvable in a reasonable amount of time. When you take that approach, it's safe to assume there must be some trick or non-linear thinking, as the answer confirms. $\endgroup$ – Todd Wilcox Jan 22 '18 at 21:34
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    $\begingroup$ Possible duplicate of How to factorise large number without calculator? $\endgroup$ – Rob Jan 23 '18 at 10:55

The thing to notice here is that 7,999,973 is close to 8,000,000. In fact it is $8000000 - 27$. Both of these are perfect cubes. Differences of cubes always factor: $$a^3 - b^3 = (a-b)(a^2+ab+b^2)$$

Here we have $a=200, b=3$, so $a-b= 197$ is a factor.

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    $\begingroup$ @Bernard: Also note that this method does not prove that the factors we got are prime. $\endgroup$ – user21820 Jan 22 '18 at 9:30
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    $\begingroup$ @user21820 As clarified in a comment, the original question only required finding a prime factor, and 197 is small enough to check by hand. $\endgroup$ – Geoffrey Brent Jan 22 '18 at 10:26
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    $\begingroup$ @GeoffreyBrent: Yup I know that 197 is prime; I'm just stating for the record that this method in general does not say anything about primality. $\endgroup$ – user21820 Jan 22 '18 at 10:33
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    $\begingroup$ @Rafalon trying to divide by 7, 11 and 13 is enough (for 197). $\endgroup$ – Will Ness Jan 22 '18 at 20:02
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    $\begingroup$ @PeterH Because the original number was suspiciously close to $8\,000\,000$, with all those $9$'s, so he checked to see whether it worked, and it did. It's a standard type of problem, although in my experience, powers of $10$ are more common as starting points for these problems than numbers like eight million. $\endgroup$ – Arthur Jan 23 '18 at 9:54

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