About my MCWC training for suprise tasks of primes or factorization, I divide the primes into 2 categories:

Trivial primes (2, 3, 5, 11) and non-trivial primes(all the other):

Non-trivial primes up to 100: 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 (21 numbers in total)

Finding divisibility with ‘trivial primes’ is very easy

2: check if odd or even

3: addition of digits should have mod3=0

5: ends in 0 or 5

11: alternate digits should be 0mod11

There are 25 numbers up to 100, and 168 primes up to 1000. But with the following trick, only memorisation of 70 special composites are needed.

So, if you want know all the primes up to 1000, it’s useful to just memorise only the 70 ‘non-trivial composites’: 91, 119, 133, 161, 203, 217, 221, 247, 259, 287, 299, 301, 323, 329, 343, 371, 377, 391, 403, 413, 427, 437, 469, 481, 493, 497, 511, 527, 533, 551, 553, 559, 581, 589, 611, 623, 629, 637, 667, 679, 689, 697, 703, 707, 713, 721, 731, 749, 763, 767, 779, 791, 793, 799, 817, 833, 851, 871, 889, 893, 899, 901, 917, 923, 931, 943, 949, 959, 973, 989. (that’s only 70 numbers to remember)

which of course are

7X13=91, 7X17=119, 7X19=133, 7X23=161, 7X29=203, 7X31=217, 13X17=221, 13X19=247, 7X37=259, 7X41=287, 13X23=299, 7X43=301, 17X19=323, 7X47=329, 7X7X7=343, 7X53=371, 13X29=377, 17X23=391, 13X31=403, 7X59=413, 7X61=427, 19X23=437, 7X67=469, 13X37=481, 17X29=493, 7X71=497, 7X73=511, 17X31=527, 13X41=533, 19X29=551, 7x79=553, 13x43=559, 7x83=581, 19x31=589, 13x47=611, 7x89=623, 17x37=629, 13X7X7=637, 23X29=667, 7X97=679, 13X53=689, 17X41=697, 19X37=703, 7X101=707, 23X31=713, 7X103=721, 17X43=731, 7X107=749, 7X109=763, 13X59=767, 19X41=779, 7X113=791, 13X61=793, 17X47=799, 19X43=817, 17X7X7=833, 23X37=851, 13X67=871, 7X127=889, 19X47=893, 29X31=899, 17X53=901, 917, 13X71=923, 19X7X7=931, 7X131=943, 13X73=949, 7X137=959, 7X139=973, 23X43=989.

So all the numbers up to 1000 fall either into any of the 168 primes or the 70 non-trivial composites (see above) or the rest 762 ‘trivial composite’ numbers which fall into any of those category 0=mod2, 0=mod3, 0=mod5, 0=mod11, which like I said are calculated very easily, almost instantly, by seeing the number.

So, to sum up actually for memorization of primes up to 1000,

there 3 categories of 3-digit numbers

- The 168 primes
- The 832 non-primes.

These 832 are categorised into

- 70 difficult ’ non-trivial’ composites
- 762 easy composites ( which have either 0=mod2, 0=mod3, 0=mod5, 0=mod11). By learning those 70 difficult ’ non-trivial’ composites, makes it much easier to instantly see if a 3-digit number is prime or not.

I think most people with good mnemonics techniques in this forum will not have any difficulty memorising those 70 numbers, if they are interested in knowing if a number is prime or not.

e.g. When I see a 3-digit number and want to factorise it, I firstly check for divisibility by 2, then by 3, then by 5, then by 11.(easy to do). If nothing is found then I recall if that number is actually contained inside my 70-number ‘special composite’ list. If not, then it’s prime.

For 4-digit numbers and above, the Eratosthenes sieve is very useful, and also important to check for div(n) only until root(n), e.g. if you see 2501 you only check 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.

In MCWC-2010 we were asked about 5-digit number factorisation in one of the surprise tasks:

http://www.recordholders.org/downloads/worldcup/tasks-2010/tasks2010.pdf

I did not know many factorization methods back then so I didn’t do my best in that competition. Also it was one the last tasks during MCWC so I was somewhat tired to check for so many divisibility criteria. The surprise task was 10 minutes for about 20 numbers. So, we were supposed to factorise each of the 5-digit numbers in about 30 seconds probably. That was a tough nut to crack. Anyway. Jerry Newport from Tuckson,USA won that primes’ task. I think Mr. Newport and Mr. Bouman have memorised all the primes up to 10,000, which of course helps a lot.

Also, recently my friend Robert Fountain and his UK team won the Chinese Brain show when he identified a 5-digit code that was a prime number. That was great feat indeed. Nice to see that this show has probably seen by millions of people in China already.

Nodas

P.S (Nov.'15):Broken link of MCWC pdf updated,due to new server transfer