Multiples, Factors & Primes

Factors: A factor is an integer that divides into another integer exactly, leaving no remainder. For any integer $n$, the set of factors is finite and always includes $1$ and $n$ itself.

Multiples: A multiple is the product of a given integer and any other whole number. Unlike factors, the set of multiples for any non-zero integer is infinite, following a linear progression ($n, 2n, 3n, ...$).

Divisibility: An integer $a$ is divisible by $b$ if the operation $a \div b$ results in an integer. This relationship is the foundation for identifying both factors (where $b$ is the factor) and multiples (where $a$ is the multiple).

Fundamental Theorem of Arithmetic: This principle states that every integer greater than $1$ is either a prime number or can be represented as a unique product of prime numbers, regardless of the order of the factors. This 'prime signature' acts like a mathematical DNA for the number.

Factor Trees: This visual method involves splitting a composite number into two factors and continuing the process until only prime numbers remain at the 'leaves' of the tree. It is a reliable way to ensure no factors are missed during decomposition.

Index Notation: To simplify the representation of prime factorization, repeated prime factors are written using exponents. For example, the decomposition of a number into $2 \times 2 \times 2 \times 3$ is expressed as $2^3 \times 3$.

Highest Common Factor (HCF): Also known as the Greatest Common Divisor (GCD), the HCF is the largest integer that divides two or more numbers without a remainder. It is found by identifying the intersection of the sets of prime factors of the numbers involved.

Lowest Common Multiple (LCM): The LCM is the smallest positive integer that is a multiple of two or more numbers. It represents the first point at which the 'cycles' of different numbers coincide.

Venn Diagram Method: By placing the prime factors of two numbers into a Venn diagram, the HCF is calculated by multiplying the factors in the intersection (overlap), while the LCM is calculated by multiplying all factors present in the entire diagram.

Check for 1 and 2: Always remember that $1$ is not a prime number and $2$ is the only even prime. Many exam questions rely on these specific properties to create 'trick' scenarios.

Reasonability Check: When calculating HCF and LCM, ensure the HCF is not larger than the smallest number in the set, and the LCM is not smaller than the largest number. If these conditions aren't met, a calculation error has likely occurred.

Keywords in Word Problems: Look for words like 'maximum', 'greatest', or 'equal groups' to signal an HCF problem. Conversely, words like 'minimum', 'next time', or 'simultaneously' usually indicate an LCM problem.

Systematic Listing: For small numbers, listing factors or multiples is often faster and less prone to error than prime factorization. However, for numbers greater than 100, prime factorization is the standard, more reliable method.