H.3 Identify multiples of a given number
What is a multiple?
A multiple is the result of multiplying a number by a whole number. Whole numbers are 0, 1, 2, 3, 4, and so on. If you can say a number equals this number times another whole number, then it is a multiple.
- Multiples of 3: 0, 3, 6, 9, 12 ... (because 3 × 0 = 0, 3 × 1 = 3, 3 × 2 = 6 ...)
- Multiples of 5: 0, 5, 10, 15, 20 ... (because 5 × 0 = 0, 5 × 1 = 5, 5 × 2 = 10 ...)
- Is 24 a multiple of 4? Yes, because 4 × 6 = 24.
- Is 30 a multiple of 7? No, because there is no whole number that equals 30 when multiplied by 7.
Every number has an endless list of multiples. You can keep multiplying by 1, 2, 3, 4, and so on, and you will never run out of multiples.
Zero is a multiple of every number
Because zero equals any number times zero (for example, 7 × 0 = 0), zero is considered a multiple of every whole number. This is an important rule in mathematics.
- 0 is a multiple of 2 (2 × 0 = 0).
- 0 is a multiple of 15 (15 × 0 = 0).
- 0 is a multiple of 100 (100 × 0 = 0).
Sometimes we focus only on nonzero multiples, but remembering that zero is a multiple helps us understand patterns in math.
How to find multiples of a given number
To find multiples of a number, simply skip-count by that number. Start with the number itself, then add the number again, and again. This is the same as multiplying the number by 1, 2, 3, 4, and so on.
- 6 × 1 = 6
- 6 × 2 = 12
- 6 × 3 = 18
- 6 × 4 = 24
- 6 × 5 = 30
- The list continues: 36, 42, 48, 54, 60 ...
- 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108 ...
A quick way to check if a larger number is a multiple of a smaller number is to perform division. If there is no remainder, then it is a multiple.
Multiples are not the same as factors
It is easy to confuse multiples and factors, but they are different. Multiples are what you get after multiplying a number by whole numbers. Factors are numbers that divide evenly into a given number.
- Multiples of 12: 12, 24, 36, 48, 60, 72 ... (these are larger or equal to 12).
- Factors of 12: 1, 2, 3, 4, 6, 12 (these are smaller or equal to 12 and divide 12 evenly).
Think of it this way: factors build a number; multiples are built from a number.
What are common multiples?
A common multiple is a number that is a multiple of two or more given numbers. In other words, it appears in the multiplication lists of each of those numbers.
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36 ...
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 ...
- Common multiples: 12, 24, 36 ... (they are in both lists).
The smallest of the common multiples (not counting zero) is called the least common multiple (LCM). In this example, the LCM of 3 and 4 is 12.
Using multiples in real-world situations
Knowing how to find multiples helps us solve many everyday problems, such as arranging items into equal groups, predicting events that repeat, and working with time or money.
Maria is making goody bags for a party. She has 24 stickers and wants to put the same number of stickers in each bag without any leftovers. She can put the stickers into 2 bags (12 each), 3 bags (8 each), 4 bags (6 each), 6 bags (4 each), 8 bags (3 each), or 12 bags (2 each). The number of bags must be a factor of 24 because 24 must be divided evenly. In other words, 24 is a multiple of the number of bags. This idea connects factors, multiples, and division.
A bus comes every 8 minutes. A train comes every 12 minutes. If they both arrive at the station right now, when will they arrive together again? You are looking for a common multiple of 8 and 12. The multiples of 8: 8, 16, 24, 32, 40, 48 ... Multiples of 12: 12, 24, 36, 48, 60 ... The first common multiple is 24. They will arrive together again in 24 minutes.
Real-world problems often ask for the least common multiple because it is the next time events happen at the same time.
Quick tricks for identifying multiples
For some numbers, you can use divisibility rules to quickly decide if a number is a multiple. These are shortcuts that save time.
- Multiple of 2: The number ends in 0, 2, 4, 6, or 8 (even numbers).
- Multiple of 3: The sum of the digits is a multiple of 3. Example: 111 → 1+1+1 = 3, so 111 is a multiple of 3.
- Multiple of 4: The last two digits form a number that is a multiple of 4. Example: 732 → last two digits 32, and 32 is a multiple of 4, so 732 is a multiple of 4.
- Multiple of 5: The number ends in 0 or 5.
- Multiple of 6: The number is even and a multiple of 3 (rules for 2 and 3 both apply).
- Multiple of 9: The sum of the digits is a multiple of 9. Example: 4,365 → 4+3+6+5 = 18, and 18 is a multiple of 9, so 4,365 is a multiple of 9.
- Multiple of 10: The number ends in 0.
These rules work because of the way our base-10 number system works. Practice them, and you will be able to spot multiples faster.
Understanding the language: "multiple of" and "divides"
When we say "a is a multiple of b," we mean there exists a whole number n such that a = b × n. This is the same as saying "b divides a" or "b is a factor of a." Learning this vocabulary helps you understand word problems.
- This is true because 7 × 6 = 42.
- We can also say "7 divides 42" or "7 is a factor of 42."
- This is false because 8 × 6 = 48 and 8 × 7 = 56. There is no whole number that gives 50.
- We can also say "8 does not divide 50" or "8 is not a factor of 50."
When a number is a multiple of another, the division results in a whole number with nothing left over.
Practice identifying multiples
The best way to master multiples is to practice. Here are some examples to think about. Cover the answers and try them yourself.
Think: 8 × 7 = 56. Yes, 56 is a multiple of 8.
Think: Multiples of 5 end in 0 or 5. 63 ends in 3, so no. Also, 5 × 12 = 60 and 5 × 13 = 65, so no.
Think: 12 × 12 = 144. Yes.
7, 14, 21, 28, 35.
16 (yes), 22 (no), 28 (yes), 34 (no), 40 (yes).
If you are ever unsure, write out the multiplication table for the number. That list contains all its multiples.
Multiples create patterns
Multiples of a number form a skip-counting pattern. Recognizing these patterns helps with mental math and understanding number relationships.
4, 8, 12, 16, 20, 24, 28, 32, 36, 40 ... The ones digit cycles through 4, 8, 2, 6, 0 over and over.
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121 ... Notice the repeating digits.
Look for patterns in the last digit or the sum of digits. These patterns come from the rules of multiplication.
Watch out for these common mistakes
Even fourth graders who understand multiples can make small errors. Being aware of these mistakes will help you avoid them.
Fact: Every number is a multiple of itself because number × 1 = number. So 12 is a multiple of 12.
Fact: Zero is a multiple of every number, but sometimes problems ask for nonzero multiples.
Fact: Factors are smaller or equal; multiples are larger or equal. For 6, factors are 1,2,3,6. Multiples are 6,12,18,24...
Fact: Every whole number is either a multiple of a given number or has a remainder when divided.
Double-check your work by doing the inverse operation. If you think 36 is a multiple of 9, divide 36 by 9. If you get 4 with no remainder, you are correct.
Key ideas to remember
Let's review the most important points about multiples.
- A multiple is the product of a number and a whole number.
- Zero is a multiple of every number.
- To find multiples, multiply the number by 0, 1, 2, 3, and so on.
- Multiples of a number are endless.
- A common multiple is a multiple shared by two or more numbers.
- Divisibility rules help you identify multiples quickly.
- Do not mix up multiples with factors.
Keep practicing with different numbers. The more you work with multiples, the more natural it becomes.
Common Core alignment: CCSS.MATH.CONTENT.4.OA.B.4 – Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.B.4. All content is 100% free; use it for whole-class instruction, independent study and practice, or homework. The material emphasizes conceptual understanding of multiples, common multiples, and the relationship between multiplication and division.