Multiples of 3

1. What Are Multiples of 3?

In mathematics, a multiple of 3 is any integer that can be expressed as 3×n, where n is a whole number (0, 1, 2, 3, …). When you multiply 3 by any non-negative integer, the product is a multiple of 3.

Multiples of 3 are part of the infinite set of numbers that follow a fixed pattern—each subsequent multiple increases by 3. Unlike factors (which divide a number evenly), multiples are the “products” you get when 3 is multiplied by other integers.

Basic Examples of Multiples of 3

  • When n=0: 3×0=0 (0 is a multiple of every integer, including 3)
  • When n=1: 3×1=3
  • When n=2: 3×2=6
  • When n=5: 3×5=15
  • When n=10: 3×10=30

A Quick List of Multiples of 3 (0–100)

0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99

2. How to Identify Multiples of 3 (The Divisibility Rule)

The easiest way to check if a number is a multiple of 3—without multiplying or dividing—is to use the divisibility rule for 3:

A number is a multiple of 3 if the sum of its digits is divisible by 3.

This rule works for numbers of any length, whether they are 2-digit, 5-digit, or even larger values.

Step-by-Step Application of the Rule

  1. Take any integer (positive or negative—negative multiples like -3, -6 also exist).
  2. Add up all of its digits.
  3. If the sum is divisible by 3, the original number is a multiple of 3. If not, it is not.

Examples of the Divisibility Rule in Action

  • Number: 126, Sum of digits: 1+2+6=99 is divisible by 3 → 126 is a multiple of 3 (3×42=126)
  • Number: 571, Sum of digits: 5+7+1=1313 is not divisible by 3 → 571 is not a multiple of 3
  • Number: 9,873, Sum of digits: 9+8+7+3=2727 is divisible by 3 → 9,873 is a multiple of 3 (3×3,291=9,873)
  • Negative Number: -45, Sum of digits (ignore the negative sign): 4+5=99 is divisible by 3 → -45 is a multiple of 3 (3×−15=−45)

3. Key Properties of Multiples of 3

Understanding the properties of multiples of 3 helps you solve math problems faster and recognize number patterns:

  1. Infinite Set: There are infinitely many multiples of 3—you can keep multiplying 3 by larger and larger integers forever.
  2. Inclusion of Zero: 0 is a multiple of 3 (3×0=0). This is true for all integers.
  3. Even and Odd Multiples: Multiples of 3 can be even or odd, depending on the value of n:
    • If n is even: 3×even=even multiple (e.g., 3×4=12)
    • If n is odd: 3×odd=odd multiple (e.g., 3×5=15)
  4. Sum and Difference Properties:
    • The sum of two multiples of 3 is also a multiple of 3. Example: 12+15=27 (27 is 3×9)
    • The difference of two multiples of 3 is also a multiple of 3. Example: 21−12=9 (9 is 3×3)
  5. Product Property: The product of a multiple of 3 and any integer is a multiple of 3. Example: 18×7=126 (126 is 3×42)
  6. Relationship to Multiples of 6 and 9:
    • All multiples of 6 are multiples of 3 (since 6 = 2×3), but not all multiples of 3 are multiples of 6 (e.g., 9 is a multiple of 3 but not 6).
    • All multiples of 9 are multiples of 3 (since 9 = 3×3), but not all multiples of 3 are multiples of 9 (e.g., 6 is a multiple of 3 but not 9).

4. How to Find Multiples of 3 (2 Simple Methods)

Method 1: Multiplication (The Direct Method)

To find the first k multiples of 3, multiply 3 by the first k whole numbers (0, 1, 2, …, k−1).

  • Example: Find the first 10 multiples of 3:3×0=0; 3×1=3; 3×2=6; …; 3×9=27Result: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27

Method 2: Skip Counting (The Sequential Method)

Skip counting by 3 is a quick way to list multiples of 3 without multiplication:

  • Start at 0, then add 3 repeatedly: 0 → 3 → 6 → 9 → 12 → 15 → …
  • This method is perfect for young learners building number sense.

5. Real-Life Applications of Multiples of 3

Multiples of 3 are not just abstract math concepts—they show up in everyday life:

  • Time Management: Hours are divided into 60 minutes, and minutes into 60 seconds—both 60 are multiples of 3. Additionally, many work shifts or class periods are 3 hours long.
  • Cooking & Baking: Recipes often call for ingredients in multiples of 3 (e.g., 3 cups of flour, 6 tablespoons of sugar) for even portioning.
  • Sports & Games: Many team sports have 3-point scoring plays (e.g., basketball 3-pointers). Board games may use 3-sided dice or require players to move 3 spaces.
  • Design & Architecture: Patterns with 3-fold symmetry (e.g., triangular tiles, 3-column layouts) rely on multiples of 3 for balance.
  • Education: Teachers often group students into teams of 3 for collaborative learning—this ensures equal distribution of work.

6. Practice Problems: Test Your Knowledge of Multiples of 3

  1. Is 237 a multiple of 3? (Sum of digits: 2+3+7=12 → Yes)
  2. List the multiples of 3 between 50 and 70. (51, 54, 57, 60, 63, 66, 69)
  3. Find the 15th multiple of 3 (starting from n=1). (3×15=45)
  4. Is the sum of 42 and 57 a multiple of 3? (42+57=99 → Sum of digits 18 → Yes)

7. Frequently Asked Questions (FAQ) About Multiples of 3

Q1: What is the definition of a multiple of 3?

A1: A multiple of 3 is any integer that can be written as 3×n, where n is a whole number (0, 1, 2, 3, …). Examples include 0, 3, 6, 9, 12, etc.

Q2: What is the divisibility rule for multiples of 3?

A2: A number is a multiple of 3 if the sum of its digits is divisible by 3. For example, 147 is a multiple of 3 because 1+4+7=12, and 12 is divisible by 3.

Q3: Is 0 a multiple of 3?

A3: Yes, 0 is a multiple of every integer, including 3. This is because 3×0=0.

Q4: Are there negative multiples of 3?

A4: Yes. Negative multiples of 3 are the result of multiplying 3 by negative integers (e.g., 3×−1=−3, 3×−2=−6).

Q5: What is the first 10 multiples of 3?

A5: The first 10 multiples of 3 (starting from n=0) are: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27.

Q6: How do you find multiples of 3 quickly?

A6: Use the divisibility rule (sum of digits divisible by 3) or skip count by 3 (0, 3, 6, 9, …). Both methods work for numbers of any size.

Q7: Is 9 a multiple of 3?

A7: Yes. 3×3=9, so 9 is the third multiple of 3 (when n=3).

Q8: Are all multiples of 3 odd?

A8: No. Multiples of 3 can be even or odd. For example, 6 (even) and 9 (odd) are both multiples of 3. The parity depends on the value of n: if n is even, the multiple is even; if n is odd, the multiple is odd.

Q9: What is the difference between factors of 3 and multiples of 3?

A9: Factors of 3 are numbers that divide 3 evenly: 1 and 3. Multiples of 3 are numbers that 3 divides evenly: 0, 3, 6, 9, etc. Factors are finite, while multiples are infinite.

Q10: Is 100 a multiple of 3?

A10: No. The sum of the digits of 100 is 1+0+0=1, which is not divisible by 3. The closest multiples of 3 to 100 are 99 and 102.

Q11: Are all multiples of 6 also multiples of 3?

A11: Yes. Since 6 = 2×3, any multiple of 6 is 2×3×n=3×(2n), which is a multiple of 3. For example, 12 is a multiple of 6 and 3.

Q12: Are all multiples of 3 also multiples of 9?

A12: No. For example, 3, 6, 12, and 15 are multiples of 3 but not of 9. Only multiples of 3 where n is a multiple of 3 are multiples of 9 (e.g., 3×3=9, 3×6=18).

Q13: How many multiples of 3 are there between 1 and 100?

A13: There are 33 multiples of 3 between 1 and 100: 3, 6, 9, …, 99. You can calculate this by dividing 100 by 3 and taking the integer part (100÷3=33.33 → 33).

Q14: What is the sum of the first 5 multiples of 3 (n=1 to 5)?

A14: The first 5 multiples of 3 are 3, 6, 9, 12, 15. Their sum is 3+6+9+12+15=45.

Q15: Can the sum of two non-multiples of 3 be a multiple of 3?

A15: Yes. For example, 4 and 5 are not multiples of 3, but 4+5=9, which is a multiple of 3.

Q16: What is the 100th multiple of 3?

A16: The 100th multiple of 3 is 3×100=300.

Q17: How do you teach multiples of 3 to kids?

A17: Use skip counting (0, 3, 6, …), songs, or manipulatives (e.g., grouping blocks into sets of 3). The divisibility rule is also easy to teach with simple numbers first.

Q18: Is 3 a multiple of itself?

A18: Yes. Every number is a multiple of itself. For 3, this is 3×1=3.

Q19: What is the least common multiple (LCM) of 3 and 5?

A19: The LCM of 3 and 5 is 15, which is the smallest number that is a multiple of both 3 and 5.

Q20: Are multiples of 3 used in algebra?

A20: Yes. In algebra, multiples of 3 are used to factor expressions (e.g., 3x+6=3(x+2)) and solve equations (e.g., finding x where 3x=21).

Q21: How do you check if a large number (e.g., 123,456) is a multiple of 3?

A21: Use the divisibility rule: sum the digits (1+2+3+4+5+6=21). Since 21 is divisible by 3, 123,456 is a multiple of 3.

Q22: What is the relationship between multiples of 3 and the number 12?

A22: 12 is a multiple of 3 (3×4=12), and all multiples of 12 are also multiples of 3. For example, 24 is a multiple of 12 and 3.

Q23: Can a decimal number be a multiple of 3?

A23: No. By definition, multiples are integers. Decimal numbers like 1.5 or 4.5 are not considered multiples of 3.

Q24: Is the difference between two multiples of 3 always a multiple of 3?

A24: Yes. Let the two multiples be 3a and 3b. Their difference is 3a−3b=3(a−b), which is a multiple of 3.

Q25: What is the greatest common multiple of 3 and 9?

A25: There is no greatest common multiple—multiples are infinite. The greatest common factor (GCF) of 3 and 9 is 3, and the least common multiple (LCM) is 9.

Q26: How do multiples of 3 relate to 3D shapes?

A26: Many 3D shapes (e.g., triangular prisms, pyramids with triangular bases) have 3 sides or faces, and their measurements (e.g., edge lengths) often use multiples of 3 for symmetry.

Q27: Are multiples of 3 used in coding or programming?

A27: Yes. Programmers often use multiples of 3 to create patterns (e.g., printing every 3rd number in a loop) or optimize code for 32-bit/64-bit systems (though 3 is more common in basic algorithms).

Q28: What is the smallest positive multiple of 3?

A28: The smallest positive multiple of 3 is 3 (when n=1). The smallest non-negative multiple is 0.

Q29: How do you find multiples of 3 in a range (e.g., 200–300)?

A29: Find the first multiple of 3 ≥ 200 (201) and the last multiple of 3 ≤ 300 (300). Then list them by adding 3 repeatedly: 201, 204, 207, …, 300.

Q30: Is 1,001 a multiple of 3?

A30: No. The sum of its digits is 1+0+0+1=2, which is not divisible by 3.

Q31: Can multiples of 3 be prime numbers?

A31: Only the number 3 itself. All other multiples of 3 are composite numbers (e.g., 6, 9, 12) because they have factors other than 1 and themselves.

Q32: How do multiples of 3 help with simplifying fractions?

A32: If both the numerator and denominator of a fraction are multiples of 3, you can simplify the fraction by dividing both by 3. For example, 1512​=15÷312÷3​=54​.

Q33: What is the sum of all multiples of 3 from 1 to 99?

A33: The multiples are 3, 6, …, 99 (33 terms). The sum of an arithmetic sequence is 2n​×(first term+last term)=233​×(3+99)=1,683.

Q34: Are multiples of 3 used in music?

A34: Yes. Music often uses 3-beat measures (e.g., waltzes with 3/4 time signature), which rely on multiples of 3 for rhythm.

Q35: How do you remember multiples of 3 easily?

A35: Use the divisibility rule, skip count regularly, or memorize a catchy song (e.g., “3, 6, 9, 12—multiples of 3 are doing well!”).

8. Conclusion

Multiples of 3 are a fundamental part of number theory, with simple identification rules and a wide range of real-life applications. Whether you’re a student learning basic math, a teacher designing lessons, or someone using math in daily tasks, understanding multiples of 3 will sharpen your number sense and problem-solving skills.

From the divisibility rule to infinite sets and real-world uses, this guide covers everything you need to master multiples of 3—start practicing today and see how these numbers shape the world around you!