Numbers are the foundation of mathematics, and the concept of multiples is among the most important to understand. You are actually learning the multiples of numbers when you study multiplication tables in elementary school. In addition to making math problems easier to solve, multiples lay the foundation for more complex ideas like factors, divisibility, the least common multiple (LCM), the highest common factor (HCF), and number theory.
A multiple of an integer is the product you get when you multiply the integer by any whole number. Multiples are a type of "skip counting" by a particular number. Multiples are often used in mathematics to find patterns, solve equations, and understand the principle of divisibility.
In mathematical language, understand it in this way: If n is an integer and k is any positive, negative, or zero integer, then.
Multiple of n = n×k
In general, the numbers that arise from multiplying a whole number by positive whole numbers—1, 2, 3, 4, etc.—are referred to as multiples. Any whole number can have an infinite number of multiples because, under certain conditions, you can simply keep multiplying. Using this as a rule, let's look at the first ten whole-number multiples:
Multiples possess some simple but fundamental properties in mathematics that help us to find, classify, and solve problems regarding multiples, such as those concerning LCM, divisibility, factors, and algebra. The following are the crucial properties of multiples:
4. Multiples can be even or odd
A common multiple is a multiple of 2 or more numbers at once. What this implies is that, if you take the two different numbers and write down the list of multiples for each number, and if you find a number in the list, then that number is a common multiple. For instance:
Of all the common multiples of two or more numbers, the smallest positive common multiple is referred to as the Least Common Multiple or the LCM. The LCM is always bigger than or equal to the largest number of the ones you are working with. For instance, common multiples of 4 and 6 are 12, 24, 36... The least common multiple of 4 and 6 is 12.
Notice that if a number is a multiple of a smaller number, then the greater number itself is the LCM.
The words "factors" and "multiples" are synonymous, yet they represent two distinct mathematical concepts. Both are used heavily in number theory, divisibility, and problem-solving. Below are the key differences between factors and multiples.
Problem 1: A teacher has 36 chocolates. Can she distribute them equally amongst nine students?
Solution: As per the question, there are 36 chocolates in total. Now, if we divide 36 by 9, we will have;
36 ÷ 9 = 4 (no remainder)
Thus, 36 is divisible by 9, and thus, each student will receive four chocolates.
Problem 2: A traffic light changes to green every 30 seconds, and a second one changes to green every 45 seconds. If both of them change to green simultaneously at 10:00 AM, after how many seconds will they change to green again simultaneously?
Solution: As per the question, we have to find the minimum time taken by the traffic light to turn green together. So in order to find it, we have to find the LCM of the duration of traffic light 1, i.e., 30 seconds, and traffic light 2, i.e., 45 seconds.
Multiples of 30 = 30, 60, 90, 120, 150, …
Multiples of 45 = 45, 90, 135, 180, …
Common multiples = 90, 180, …
LCM = 90 seconds.
Therefore, they will turn green together once more after 90 seconds (1 minute and 30 seconds).
Problem 3: Write the first five multiples of 11.
Solution: Multiples of a number are found by multiplying it by 1, 2, 3, 4, 5… thus:
11 × 1 = 11
11 × 2 = 22
11 × 3 = 33
11 × 4 = 44
11 × 5 = 55
(Session 2026 - 27)