# Prime numbers

Prime numbers are the natural numbers that have just two different dividers: 1 and himself.

Examples:

1) 2 have only the dividers 1 and 2, therefore 2 It's a prime number.
2) 17 have only the dividers 1 and 17, therefore 17 It's a prime number.
3) 10 have the dividers 1, 2, 5 and 10, therefore 10 no It's a prime number.

1 is not a prime number, because he has only one divider that is himself.
2 is the only prime number that is even.

Numbers that have more than two dividers are called compound numbers.
Example: 15 has more than two dividers => 15 is a compound number.

## Recognition of a prime number

To know if a number is prime, we divide that number by prime numbers 2, 3, 5, 7, 11, etc., until we have:
- or a division with zero remainder (and in this case the number not cousin),
- or a division with lower quotient that the divisor and the nonzero rest. In this case the number it's cousin.

Examples:

1) The number 161:

• not even, so not divisible by 2;
• 1 + 6 + 1 = 8, so it is not divisible by 3;
• does not end in 0 or 5, so it is not divisible by 5;
• by 7: 161/7 = 23, with zero remainder, so 161 is divisible by 7, and therefore no It's a prime number.

2) The number 113:

• not even, so not divisible by 2;
• 1 + 1 + 3 = 5, so it is not divisible by 3;
• does not end in 0 or 5, so it is not divisible by 5;
• by 7: 113/7 = 16, with remainder 1. The quotient (16) is still greater than the divisor (7).
• by 11: 113/11 = 10, with remainder 3. The quotient (10) is smaller than divisor (11), and furthermore the remainder is nonzero (the remainder is 3), so 113 is a prime number.
Next: Prime Fact Decomposition