Factors and Multiples

For two integers $P$ and $Q$, if $\frac{P}{Q}$ is an integer, then $Q$ divides $P$ exactly: $Q$ is a factor of $P$, and $P$ is a multiple of $Q$. Every whole number has a limited list of factors (including $1$ and itself) but an endless list of multiples.

To list the factors of a number, find every pair that multiplies to it. For $24$: $1 \times 24$, $2 \times 12$, $3 \times 8$, $4 \times 6$. So the factors of $24$ are $1, 2, 3, 4, 6, 8, 12, 24$.

The greatest common factor (GCF) of two numbers is the largest factor they share. The factors of $12$ are $1, 2, 3, 4, 6, 12$ and of $18$ are $1, 2, 3, 6, 9, 18$; the largest shared one is $6$.

The least common multiple (LCM) is the smallest positive multiple they share. Multiples of $4$ are $4, 8, 12, 16, \dots$ and of $6$ are $6, 12, 18, \dots$; the smallest shared one is $12$.

Is $4$ a factor of $18$?
Check $\frac{18}{4} = 4.5$, which is not an integer, so $4$ is not a factor of $18$.
By contrast $\frac{18}{3} = 6$ is an integer, so $3$ is a factor of $18$ and $18$ is a multiple of $3$.

What is the greatest common factor of $16$ and $24$?
Factors of $16$: $1, 2, 4, 8, 16$. Factors of $24$: $1, 2, 3, 4, 6, 8, 12, 24$.
The largest factor they share is $8$.

What is the least common multiple of $6$ and $8$?
Multiples of $6$: $6, 12, 18, 24, \dots$ Multiples of $8$: $8, 16, 24, \dots$
The smallest multiple they share is $24$.