Number theory introduces fundamental concepts of prime numbers and divisibility, which are tightly interrelated: any natural (whole) number is defined as prime number (or simply prime) if it’s evenly divided only by 1 and the number itself (these two are called trivial divisors). The rest of natural numbers, which are not primes, are called composite numbers; they contain at least one non-trivial divisor.
Primality test is a mathematical procedure intended to prove that the natural number is either prime, or composite. Primality test could be performed by factoring the natural numbers, i.e. presenting the number as a product of smaller natural numbers called factors or divisors; the latter two terms could be used interchangeably.
Primality test as well as factoring of big natural numbers is not a trivial task and could require substantial about of time/labor. In certain cases, simple divisibility test could dramatically reduce the amount of calculations, required for the primality test.
Divisibility rules (also called divisibility tests) are kind of numeric shortcuts, intended to determine in a simple way if the natural number is evenly divisible by another number; if so, it’s not a prime, but a composite number. The most commonly used divisibility rules, applicable to any natural number, are listed below:
- Divisibility rule for 2 is the simplest one; any natural even number (i.e. the number ending in 0,2,4,6 or 8) is divisible by 2.
- Divisibility rule for 3 states, that if the sum of all digits in the natural number is evenly divisible by 3, so is the number itself.
- Divisibility rule for 4 is slightly more difficult: it states that any even number with last 2 digits evenly divisible by 4 is also evenly divisible by 4.
- Divisibility rule for 5 states, that any natural number ending in 0 or 5 is evenly divisible by 5.
- Divisibility rule for 6 states that any natural number evenly divisible by 2 and 3 is also divisible by 6.
- Divisibility rule for 8 states that if the last three digits of the natural number are evenly divided by 8 so is the entire number.
- Divisibility rule for 10 is the most trivial one: any natural number ending in 0 is evenly divisible by 10.
Other divisibility rules exist, though they are less straightforward and require more complex calculations, which limited their usefulness. Instead, it is recommended to perform the primality test and even full prime factoring by using online prime factoring calculator. Users of Apple iPod Touch™, iPhone™, Microsoft Zune™ HD could benefit from using the mobile prime factoring calculator, optimized for reduced screen-size portable devices.
Recommended reading on Examiner.com
- Online prime factoring calculator
- Concise introduction to number theory: rational, irrational and transcendental numbers
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