What is the best way to test primality of this number $123456789101112131415161718192021222324252627282931$ for high school level? I found that the number : $$123456789101112131415161718192021222324252627282931$$ which is prime number , I want to know if there is any simple method to show students its primality ?
 A: For high school level, showing non-primality is fair game, provided simple divisibility criteria can resolve it. 

Showing primality for a number such as the one you posted is definitely not fair game.

One guideline would be: If the teacher can't show it easily, then it's not appropriate for students.
A: The number is a prime. This cannot be found out in reasonable time without a computer.
A: This isn't practical. Without knowing every prime below about $10^{25}$ you can't do simple divisibility testing (even if you could do it all before the universe ends). Even sieving with a $$6k\pm1$$ sieve adapted from the sieve of sundaram you would either need to type out more than $10^{24}$ numbers and sieve them as you go, just to get base primes. Primorial base change only helps to a point.  Beyond which a primality check of some form must be known. Preferable a fast one.  
For a comparison , I wrote out all numbers up to 800 ( might go a bit further) but even if I do a sieve based on form, I'll only get all primes under about 4900 back. it's taken me about a week on and off. Yours would take a lifetime.  Have fun trying to teach primality checks.  even mod 30 we have 6 equations for it's mod 30 class to check. 
A: It would be difficult for high-school, and impossible without computers. That being said, $with$ computers, you can apply the Lucas primality test with the number $a=2$. Note that the factorization of $n-1$ is
$$n-1 = 2×3×5×13×49269439\times 370677592383442753 \times 17333107067824345178861$$
