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It is useful to recognize prime numbers, especially those between 1-100. Knowing them helps you to efficiently tackle many mathematics problems involving factorization, highest/greatest common factor, lowest common multiple, divisibility test, expression simplification, modulo arithmetic, Diophantine equation, etc.

How do you memorize or recognize prime numbers from 1-1000? Here are some tricks (for human brains!) I found out. Tell me if you have better methods (for human brains but not for computers!).

Level 1: Prime Numbers within 1-10

It is pretty easy to memorize the only 4 prime numbers here: 2, 3, 5, 7. You MUST know them well!

Level 2: Prime Numbers within 11-100

A few facts that you should know before you proceed further:

There are exactly 25 prime numbers within 1-100.
A number is divisible by 2 if and only if its last digit is even (0, 2, 4, 6, or 8).
A number is divisible by 3 if and only if the sum of its digits is divisible by 3.
A number is divisible by 5 if and only if its last digit is either 0 or 5.
If a number N is not a prime number, then it must be divisible by a prime number that is less than or equal to the square root of N.

So, to check whether a number is prime within 11-100:

If a number does NOT end with 1, 3, 7, or 9, then it is definitely NOT a prime number. This tackles all numbers that are divisible by 2 or 5.
Sum up its digits to check if it is divisible by 3.
Check if the number is divisible by 7. You should have memorized the multiplication table up to either 10x10 or 12x12. So, if the number is less than 70, you can check quickly whether it can be produced with 7 times some integer. If the number is more than 70 (but less than 100), first minus it by 70, and then check as mentioned above. For example, take 97. Minus it by 70, we get 27, in which you can't get it using 7 times some integer. So 97 is not divisible by 7. A few tests do exist for divisibility by 7, but I find them less efficient when dealing with numbers less than 100. Anyway, you are encouraged to learn more about divisibility rules.

This is the list of all 21 prime numbers within 11-100:

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

I have purposely arranged these numbers as shown so that you may see some patterns to help you memorizing them. Memorize them if you can!

Level 3: Prime Numbers within 101-1000

This is getting tough! You can stop here after memorizing all 25 prime numbers that are less than 100.

However, if you are really keen, you may be interested to know a few things:

There are exactly 168 prime numbers within 1-1000.
To save your effort, first estimate the square root of the given number to find out what divisibility tests you need to use. For example, to check for a number which is less than 400, you only need to try to divide it by the prime numbers up to 20, which is the square root of 400.
It helps to learn more about various divisibility rules.

Here's the list of all prime numbers within 1-1000:

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
101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293
307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397
401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499
503 509 521 523 541 547 557 563 569 571 577 587 593 599
601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691
701 709 719 727 733 739 743 751 757 761 769 773 787 797
809 811 821 823 827 829 839 853 857 859 863 877 881 883 887
907 911 919 929 937 941 947 953 967 971 977 983 991 997