Key Points

A factor of a number divides it exactly, while a multiple is the result of multiplying the number by an integer. For example, 4 is a factor of 12, and 12 is a multiple of 4.

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself. Examples include 2, 3, 5, 7, 11, and 13.

A composite number is a natural number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 10.

The number 1 is unique because it is neither a prime number nor a composite number. It has only one factor, which is 1.

The number 2 is the smallest prime number and the only even prime number. All other even numbers are composite because they are divisible by 2.

Every composite number can be written as a unique product of prime numbers. This is called its prime factorisation. For example, the prime factorisation of 84 is $2 \times 2 \times 3 \times 7$.

Common factors are numbers that are factors of two or more different numbers. For example, the common factors of 20 and 28 are 1, 2, and 4.

Common multiples are numbers that are multiples of two or more different numbers. For example, the common multiples of 4 and 6 are 12, 24, 36, and so on.

Two numbers are called co-prime if their only common factor is 1. For example, 18 and 35 are co-prime.

To check if two numbers are co-prime, find their prime factorisations. If there are no common prime factors, the numbers are co-prime. For example, $80 = 2^4 \times 5$ and $63 = 3^2 \times 7$ have no common prime factors, so they are co-prime.

A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).

A number is divisible by 5 if its last digit is 0 or 5.

A number is divisible by 10 if its last digit is 0.

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For example, 8536 is divisible by 4 because 36 is divisible by 4.

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. For example, 14560 is divisible by 8 because 560 is divisible by 8.

A number is divisible by another if the prime factorisation of the second number is completely contained within the prime factorisation of the first. For example, $168 = 2^3 \times 3 \times 7$ is divisible by $12 = 2^2 \times 3$.

A perfect number is a positive integer where the sum of its factors (including itself) is equal to twice the number. For example, 6 is a perfect number because its factors are 1, 2, 3, 6, and $1+2+3+6 = 12 = 2 \times 6$.

Twin primes are pairs of prime numbers that have a difference of 2. For example, (3, 5), (11, 13), and (17, 19) are twin primes.