The First Twenty Five Primes

Every prime number is a positive integer that is divisible by 1 and itself only. There are infinitely many prime numbers. Prime numbers are peculiar numbers which do not have a common predictable property other than that of the indivisibility. Because they are indivisible, the fundamental theorem of arithmetic is that they are the building blocks of numbers. That is to say that any positive integer greater than 1 can be written uniquely as the product of prime numbers.

There are 25 primes below 100. This is a method to remember all the first 25 primes.
We begin with 25 itself.

The first and third primes are 2 and 5 respectively. They are prime numbers but no other prime number would have 2 and 5 as its end digits
Now, 5+2 and 5-2 give the other two single digits primes. i.e., 7 and 3.
Hence, we have all the single digit primes as 2, 3, 5 and 7.

All primes with at least two digits would end only in 1, 3, 7 or 9.

We list all the two digit primes that end in 1.

Wen use 2 and 5 once again for that.
Splitting 2, we get the first two digit prime, 11.
Then, 1+2=3, 1+3=4, 1+5=6 and 1+6=7 are the first digits of the remaining primes.
Hence, the two digit primes are 11, 31, 41, 61 and 71. Thus we get all primes that end in 1.

A total of 9 primes, we have found out so far.

Now, we need to list all the primes that end in 3.

Adding the digits of the primes that end in 1, we get 1+1=2, 3+1=4, 4+1=5, 6+1=7 and 7+1=8.
Appending 3 to the right of the above numbers, we get the two digit primes that end in 3.
They are 23, 43, 53, 73 and 83. Hence, excluding 3, we got 13 primes. Of course, 13 is also a prime number.

What left are the primes that end in 7 or 9. But, this is very easy.

List all the digits in numbers from 1 to 10.
They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 and 3.
Appending 7 or 9 as the case may be(by avoiding the trivial composite numbers we get,
17, 29, 37, 47, 59, 67, 79, 89, 97, 19 and 07.
These are the 11 prime numbers that end in 7 or 9.

Hence all the prime numbers below 100 are:

1. 2 and 5
2. 11, 31, 41, 61 and 71
3. 3, 13, 23, 43, 53, 73 and 83.
4. 7, 17, 37, 47, 67 and 97
5. 19, 29, 59, 79 and 89.