The Rules of Divisibility by Certain Integers:
- An integer is divisible by 2 if the integer is even.
- An integer is divisible by 3 if the sum of the integer's digits is divisible by 3.
- An integer is divisible by 4 if the integer is divisible by 2 twice or if the last two digits are divisible by 4.
- An integer is divisible by 5 if the integer ends in 0 or 5.
- An integer is divisible by 6 if the integer is divisible by both 2 and 3.
- An integer is divisible by 8 if the integer is divisible by 2 three times or if the last three digits are divisible by 8.
- An integer is divisible by 9 if the sum of the integer's digits is divisible by 9.
- An integers is divisible by 10 if the integer ends in 0.
The Factor-Foundation Rule:
If A is a factor of B, and B is a factor of C, then A is a factor of C.
The Quotient-Reminder Theorem:
Dividend = Quotient * Divisor + Remainder
The Three Arithmetic Rules of Evens and Odds for Addition and Subtraction:
- Even +- Even = Even
- Odd +- Odd = Even
- Even +- Odd = Odd
The Three Arithmetic Rules of Evens and Odds for Multiplication:
- Even * Even = Even
- Even * Odd = Even
- Odd * Odd = Odd
The Words "Or" and "And":
"Or" means add and "And" means multiply.
All Primes up to 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, and 97.
The Greatest Common Factor:
The product of the primes in the shared area of the respective Venn diagram.
The Least Common Multiple:
The product of all primes in the respective Venn diagram.
The Formula for Counting the Number of Total Factors of a Number:
In general, if a prime factor appears to the N-th power, then there are ( N + 1 ) possibilities for the occurrence of that prime factor.
Therefore, if a number has the prime factorisation A^X * B^Y * C^Z, then the number has ( X +1 ) * ( Y + 1 ) * ( Z + 1 ) different factors.
The Prime Factorisation of a Perfect Square:
The prime factorisation of a perfect square contains only even powers of primes.
If a number's prime factorisation contains any odd powers of primes, then the number cannot be a perfect square.
The Prime Factorisation of a Perfect Cube:
If a number is a perfect cube, then it is formed by three identical sets of primes; therefore, in the factorisation of a perfect cube, all the powers of the respective primes have to be multiples of 3.
The Two Rules for Arithmetic Operations on Integers:
1. The sum of two integers is always an integer.
2. The difference of two integers is always an integer.
If there are no restrictions, the number of ways of arranging N distinct objects is N!.
The Special Property of Perfect Squares with Respect to the Number of Total Factors:
All perfect squares have an odd number of total factors, while all other non-square integers have an even number of total factors.