Multiples and Factors - Multiples and Least Common Multiples

A multiple of a number is the product obtained when that number is multiplied by a counting number like $1, 2, 3, \dots$. Visually, if you represent multiples on a number line, a multiple of $4$ is shown by taking equal jumps of $4$ units each, landing on $4, 8, 12, 16, \dots$.

Every number is a multiple of $1$ and also a multiple of itself. The smallest multiple of any number is the number itself. Unlike factors, the number of multiples for any given number is infinite because counting numbers never end.

Common Multiples are numbers that are multiples of two or more different numbers. Imagine two people jumping on a number line: person A jumps by $2$s and person B jumps by $3$s. The points where they both land (like $6, 12, 18, \dots$) are the common multiples of $2$ and $3$.

The Least Common Multiple (LCM) is the smallest number among all the common multiples of two or more numbers. On a number line, the LCM is the very first point (other than zero) where the jumps of different-sized steps meet.

To find the LCM by the Listing Method, write out the multiples of each number in a horizontal list. For example, for $3$ and $4$, multiples of $3$ are $\{3, 6, 9, 12, \dots\}$ and multiples of $4$ are $\{4, 8, 12, 16, \dots\}$. The first number to appear in both lists ($12$) is the LCM.

The Common Division Method (or Ladder Method) involves writing numbers in a row and dividing them by the smallest prime number that can divide at least one of the numbers. Visually, this looks like an 'L-shaped' table where you keep dividing until the last row consists of $1$s. The LCM is the product of all the prime divisors used.

An important property of multiples is that the LCM of two co-prime numbers (numbers that have no common factor other than $1$) is simply their product. For example, the LCM of $5$ and $7$ is $5 \times 7 = 35$.