Divisors, multiples and prime factorisation

Divisors and multiples

The divisors of $12$ are $1, 2, 3, 4, 6, 12$. Listing the pairs with product $12$ ($1 \times 12$, $2 \times 6$, $3 \times 4$) makes sure you miss none.

The multiples of $6$ are $6, 12, 18, \ldots$ The number of multiples of $k$ from $1$ to $N$ is the quotient of $N \div k$.

The largest common divisor of two numbers is their greatest common divisor (GCD); the smallest common multiple is their least common multiple (LCM).

Primes and prime factorisation

A prime has no divisors other than $1$ and itself ($2, 3, 5, 7, 11, 13, \ldots$; $1$ is not prime).

Prime factorisation writes a number as a product of primes. Divide repeatedly by small primes:

Finding the GCD and LCM

• GCD: multiply the common primes, each to the lower power: $2 \times 3 \times 7 = 42$
• LCM: multiply every prime, each to the higher power: $2^2 \times 3^2 \times 7 = 252$