Can you tell me the prime numbers up to 100?

Question

There is no formula (hence the reason they are prime), but there is a way to mathematically figure it out. Tell me your answers and how you figured them out. The best "formual" gets the points.
Hint: Stephen King wrote about it in one of teh Gunslinger Series books.

Answer 1

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

I went to the Online Encyclopedia of Integer Sequences and typed the first few primes that I'd memorized. It found the prime number sequence and I copied it.

For small primes, you can just manually run through the numbers and mentally figure out if they have any factors (Simply check for even numbers, multiples of 3 (digits add to 3), multiples of 5 (last digit 0 or 5), etc.)

You can also use the 'sieve' method to cross out all powers of 2, 3, 5, 7. That's as far as you need to go in this case because the next prime is 11 and 11² > 100.

Original list (remove 1 which isn't prime):
- 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100

Remove powers of 2 (except 2):
- 2 3 - 5 - 7 - 9 -
11 - 13 - 15 - 17 - 19 -
21 - 23 - 25 - 27 - 29 -
31 - 33 - 35 - 37 - 39 -
41 - 43 - 45 - 47 - 49 -
51 - 53 - 55 - 57 - 59 -
61 - 63 - 65 - 67 - 69 -
71 - 73 - 75 - 77 - 79 -
81 - 83 - 85 - 87 - 89 -
91 - 93 - 95 - 97 - 99 -

Remove powers of 3 (except 3):
- 2 3 - 5 - 7 - - -
11 - 13 - - - 17 - 19 -
- - 23 - 25 - - - 29 -
31 - - - 35 - 37 - - -
41 - 43 - - - 47 - 49 -
- - 53 - 55 - - - 59 -
61 - - - 65 - 67 - - -
71 - 73 - - - 77 - 79 -
- - 83 - 85 - - - 89 -
91 - - - 95 - 97 - - -

Remove powers of 5 (except 5):
- 2 3 - 5 - 7 - - -
11 - 13 - - - 17 - 19 -
- - 23 - - - - - 29 -
31 - - - - - 37 - - -
41 - 43 - - - 47 - 49 -
- - 53 - - - - - 59 -
61 - - - - - 67 - - -
71 - 73 - - - 77 - 79 -
- - 83 - - - - - 89 -
91 - - - - - 97 - - -

Remove powers of 7 (except 7):
- 2 3 - 5 - 7 - - -
11 - 13 - - - 17 - 19 -
- - 23 - - - - - 29 -
31 - - - - - 37 - - -
41 - 43 - - - 47 - - -
- - 53 - - - - - 59 -
61 - - - - - 67 - - -
71 - 73 - - - - - 79 -
- - 83 - - - - - 89 -
- - - - - - 97 - - -

That leaves the list we had before:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Answer 2

No even number is prime other than 2.
Any number that ends in 5 is not prime, other than 5 itself.
so you only have to check numbers that end in an odd digit other than 5
Digits can't add up to a number divisible by 3 or else the number is also divisible by 3 (ex: 63....6+3 = 9, 9 is divisible by 3, so 63 must be also)
Ex for the 20's: check 21, 23, 27, 29: 21(1+2 = 3) & 27(2+7 = 9) aren't prime, 23 & 29 are.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 57, 59, 61, 67, 71, 73, 79, 83, 89, 97

Answer 3

2-3-5-7-11-13-17-19-23-27-29-31-37-39-41-43-47-51-53-57-59-61-63-67-69-71-73-79-83-87-89-91-93-97-