Determining Common Multiples

A multiple of a number is what you get when you multiply that number by a whole number: \(1, 2, 3, 4, \dots\). For example, the multiples of \(4\) are \(4, 8, 12, 16, 20, 24, 28, \dots\), and the multiples of \(6\) are \(6, 12, 18, 24, 30, \dots\). A common multiple of two or more numbers is any value that shows up in every one of their multiples lists. Comparing these lists is closely related to comparing factors, which is covered in Determining Common Factors, so it helps to be comfortable with that idea first.

The most direct way to determine common multiples is to write out the multiples of each number and look for matches.

1. List several multiples of the first number.
2. List several multiples of the second number (and any additional numbers).
3. Circle or underline the values that appear in every list.

Notice that \(12\) and \(24\) appear in both rows above, so they are common multiples of \(4\) and \(6\). If you kept listing further, you would also find \(36\), \(48\), and so on, since common multiples continue forever in a predictable pattern.

Listing works well for small numbers, but it gets slow when the numbers are large or when you need several common multiples at once. In that case, breaking each number into its prime factors makes the pattern of shared multiples much easier to see. If you have not reviewed this yet, the prime factorization method lesson walks through how to break a number down into primes step by step. Once you know the prime factors, you can combine them to build multiples that both numbers share, rather than guessing and checking a long list.

Every pair (or group) of numbers has infinitely many common multiples, but the smallest one has a special name: the least common multiple, or LCM. For \(4\) and \(6\), the least common multiple is \(12\), since it is the smallest value that appears in both lists. Determining common multiples is really the first step toward finding this smallest shared value. For a full walkthrough of that process, including shortcuts using divisibility rules, see the dedicated Least Common Multiple (LCM) lesson.

Find the common multiples of \(3\) and \(5\) that are less than \(40\).

Multiples of \(3\): \(3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39\).

Multiples of \(5\): \(5, 10, 15, 20, 25, 30, 35\).

Comparing the lists, \(15\) and \(30\) appear in both, so the common multiples of \(3\) and \(5\) below \(40\) are \(15\) and \(30\).

A frequent mix-up is confusing common multiples with common factors: factors divide into a number, while multiples are what you get from multiplying it. It also helps to recognize numbers quickly using the divisibility rule of 4 when checking long lists, and understanding place value keeps you from misreading larger multiples as you count.

Related lessons

• Basic Math Help