Multiples of 11

1. What Are Multiples of 11?

In mathematics, a multiple of 11 is any integer that can be expressed as 11×n, where n is an integer (positive, negative, or zero). When 11 is multiplied by any whole number or negative integer, the resulting product belongs to the infinite set of multiples of 11.

11 is a prime number, and its multiples exhibit unique digit patterns that set them apart from multiples of smaller primes. These patterns make identifying multiples of 11 intuitive once you learn the core rules, even for large multi-digit numbers.

Basic Examples of Multiples of 11

  • When n=0: 11×0=0 (0 is a multiple of every integer)
  • When n=1: 11×1=11
  • When n=9: 11×9=99
  • When n=15: 11×15=165
  • When n=−6: 11×(−6)=−66 (negative multiples follow the same pattern as positive ones)

Quick List of Multiples of 11 (0–200)

0, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121, 132, 143, 154, 165, 176, 187, 198

2. How to Identify Multiples of 11 (The Divisibility Rule)

The divisibility rule for 11 is a bit more nuanced than rules for smaller numbers, but it is highly effective for numbers of any length. There are two complementary versions of the rule, depending on the number of digits:

Rule 1: For 2-Digit Numbers

A 2-digit number is a multiple of 11 if and only if its two digits are identical.Examples: 11 (1=1), 22 (2=2), 99 (9=9) → all multiples of 11; 12 (1≠2), 35 (3≠5) → not multiples of 11.

Rule 2: For 3-Digit and Larger Numbers

Subtract the sum of the digits in the even positions from the sum of the digits in the odd positions (counting positions from right to left, starting at 1). If the result is 0 or a multiple of 11 (including negative multiples), the number is a multiple of 11.

Step-by-Step Application

  1. Write down the number and label each digit’s position from right to left (Position 1 = rightmost digit).
  2. Calculate the sum of digits in odd positions (1, 3, 5…).
  3. Calculate the sum of digits in even positions (2, 4, 6…).
  4. Find the absolute difference between the two sums.
  5. If the difference is 0 or divisible by 11, the number is a multiple of 11.

Examples of the Rule in Action

  • Number: 132
    1. Positions (right to left): 2 (1), 3 (2), 1 (3)
    2. Sum of odd positions (1,3): 2 + 1 = 3
    3. Sum of even positions (2): 3
    4. Difference: 3−3=0
    5. 132 is a multiple of 11 (11×12=132)
  • Number: 1,210
    1. Positions (right to left): 0 (1), 1 (2), 2 (3), 1 (4)
    2. Sum of odd positions (1,3): 0 + 2 = 2
    3. Sum of even positions (2,4): 1 + 1 = 2
    4. Difference: 2−2=0
    5. 1,210 is a multiple of 11 (11×110=1210)
  • Number: 7,161
    1. Positions (right to left): 1 (1), 6 (2), 1 (3), 7 (4)
    2. Sum of odd positions (1,3): 1 + 1 = 2
    3. Sum of even positions (2,4): 6 + 7 = 13
    4. Difference: ∣2−13∣=11 (a multiple of 11)
    5. 7,161 is a multiple of 11 (11×651=7161)
  • Number: 458
    1. Positions (right to left): 8 (1), 5 (2), 4 (3)
    2. Sum of odd positions (1,3): 8 + 4 = 12
    3. Sum of even positions (2): 5
    4. Difference: 12−5=7 (not 0 or multiple of 11)
    5. 458 is not a multiple of 11

3. Key Properties of Multiples of 11

Understanding the properties of multiples of 11 helps uncover number patterns and simplify problem-solving across arithmetic, algebra, and number theory:

  1. Infinite Set: There are infinitely many multiples of 11—multiply 11 by any integer (positive, negative, zero) to generate a new multiple with no upper or lower bound.
  2. Inclusion of Zero: 0 is a multiple of 11 (11×0=0), as it is for all integers in the number system.
  3. Prime Factor Trait: Since 11 is prime, its only positive factors are 1 and 11. Every non-zero multiple of 11 has 11 as one of its prime factors.
  4. Alternating Parity: Multiples of 11 alternate between odd and even numbers, depending on the multiplier n:
    • If n is odd → multiple of 11 is odd (e.g., 11×3=33, odd)
    • If n is even → multiple of 11 is even (e.g., 11×4=44, even)
  5. Sum and Difference Properties:
    • The sum of two multiples of 11 is a multiple of 11. Example: 22+55=77 (11×7=77)
    • The difference of two multiples of 11 is a multiple of 11. Example: 121−33=88 (11×8=88)
  6. Product Property: The product of a multiple of 11 and any integer is a multiple of 11. Example: 66×5=330 (11×30=330)
  7. Relationship to Multiples of 22, 33, and 44:
    • All multiples of 22, 33, and 44 are multiples of 11 (since 22=2×11, 33=3×11, 44=4×11)
    • Not all multiples of 11 are multiples of 22 (e.g., 11, 33 are multiples of 11 but not 22)
  8. 2-Digit Pattern: All 2-digit multiples of 11 have identical digits (11, 22, …, 99).
  9. 3-Digit Pattern: For 3-digit multiples of 11, the middle digit is the sum of the first and last digits (e.g., 132: 1 + 2 = 3; 165: 1 + 5 = 6).

4. How to Find Multiples of 11 (2 Simple Methods)

Finding multiples of 11 is straightforward with two reliable methods, suitable for learners of all skill levels:

Method 1: Multiplication (Direct Calculation)

To find the first k multiples of 11, multiply 11 by the first k whole numbers (0, 1, 2, …, k−1). This method is ideal for generating specific multiples or ordered lists.

Example: Find the first 10 multiples of 11:11×0=0; 11×1=11; 11×2=22; 11×3=33; 11×4=44; 11×5=55; 11×6=66; 11×7=77; 11×8=88; 11×9=99Result: 0, 11, 22, 33, 44, 55, 66, 77, 88, 99

Method 2: Skip Counting (Sequential Listing)

Skip counting by 11 is a beginner-friendly way to list multiples of 11 without formal multiplication. It builds number fluency and reinforces the sequential pattern of 11’s multiples.

Start at 0 and add 11 repeatedly to generate the sequence:0 → 11 → 22 → 33 → 44 → 55 → 66 → …

5. Real-Life Applications of Multiples of 11

Multiples of 11 are less common in daily life than smaller multiples, but they still appear in specific systems, measurements, and industries:

  • Time & Calendars: Some cultures use 11-day cycles for traditional festivals or agricultural planning. In sports, 11-player teams (e.g., soccer, cricket) are a global standard—directly tied to the number 11.
  • Retail & Pricing: Promotional pricing sometimes uses multiples of 11 (e.g., $11, $22, $99) for psychological pricing—these numbers feel “rounded” but distinct from common 5 or 10 multiples.
  • Education & Testing: Teachers use multiples of 11 for problem-solving exercises, especially to teach divisibility rules for larger primes. Standardized tests often include questions about multiples of 11 to assess number sense.
  • Manufacturing & Packaging: Some products are packaged in multiples of 11 (e.g., 11-ounce snack bags, 22-count paper towel rolls) to differentiate from competitors using 10 or 12-packs.
  • Gaming & Scoring: Games use multiples of 11 for bonus points (e.g., 110 points for a hidden collectible) or level milestones (unlocking a reward at level 11, 22, or 33).
  • Finance & Accounting: Some loan repayment plans use 11-month terms, and small businesses may track inventory in batches of 11 for niche products.

6. Practice Problems: Test Your Knowledge of Multiples of 11

  1. Is 253 a multiple of 11? (Sum of odd positions: 3 + 2 = 5; sum of even positions: 5; difference 0 → Yes)
  2. List the multiples of 11 between 100 and 200. (110, 121, 132, 143, 154, 165, 176, 187, 198)
  3. Find the 50th multiple of 11 (n=50). (11×50=550)
  4. Is the sum of 187 and 209 a multiple of 11? (187+209=396; sum of odd positions: 6 + 3 = 9; sum of even positions: 9; difference 0 → Yes)
  5. What is the smallest multiple of 11 greater than 1,000? (1001)

7. Frequently Asked Questions (FAQ) About Multiples of 11

Q1: What is the formal definition of a multiple of 11?

A1: A multiple of 11 is any integer that can be written as 11×n, where n is an integer (positive, negative, or zero). Examples include 0, 11, 22, -11, -22, and so on.

Q2: What is the divisibility rule for multiples of 11?

A2: For 2-digit numbers: digits are identical. For 3+ digit numbers: subtract the sum of even-positioned digits from odd-positioned digits (right-to-left). If the result is 0 or a multiple of 11, the number is a multiple of 11.

Q3: Is 0 a multiple of 11?

A3: Yes. 0 is a multiple of every integer, including 11. This is because 11×0=0.

Q4: Are there negative multiples of 11?

A4: Yes. Negative multiples of 11 are created by multiplying 11 by negative integers. Examples include -11 (11×−1), -66 (11×−6), and -121 (11×−11).

Q5: What are the first 10 multiples of 11?

A5: The first 10 multiples of 11 (starting from n=0) are: 0, 11, 22, 33, 44, 55, 66, 77, 88, 99. For positive-only multiples (starting from n=1), they are: 11, 22, 33, 44, 55, 66, 77, 88, 99, 110.

Q6: How do you find multiples of 11 quickly?

A6: Use two methods: (1) Apply the divisibility rule (identical digits for 2-digit numbers; sum difference for longer numbers); (2) Skip count by 11 starting from 0 to generate new multiples.

Q7: Is 121 a multiple of 11?

A7: Yes. 11×11=121. Using the divisibility rule: sum of odd positions (1 + 1 = 2), sum of even positions (2), difference 0 → 121 is a multiple of 11.

Q8: Are all multiples of 11 odd?

A8: No. Multiples of 11 alternate between odd and even. If the multiplier is odd, the multiple is odd (e.g., 11×5=55). If the multiplier is even, the multiple is even (e.g., 11×6=66).

Q9: What is the difference between factors of 11 and multiples of 11?

A9: Factors of 11 are numbers that divide 11 evenly: 1 and 11 (since 11 is prime). Factors are finite. Multiples of 11 are numbers that 11 divides evenly: 0, 11, 22, etc. Multiples are infinite.

Q10: Is 374 a multiple of 11?

A10: Yes. Sum of odd positions (4 + 3 = 7), sum of even positions (7), difference 0 → 374 is a multiple of 11 (11×34=374).

Q11: Are all multiples of 22 also multiples of 11?

A11: Yes. Since 22=2×11, any multiple of 22 can be written as 2×11×n, which means it is automatically divisible by 11.

Q12: Are all multiples of 11 also multiples of 22?

A12: No. Only multiples of 11 with even multipliers are multiples of 22 (e.g., 22, 44). Multiples of 11 with odd multipliers (e.g., 11, 33) are not multiples of 22.

Q13: How many multiples of 11 are there between 1 and 100?

A13: There are 9 multiples of 11 between 1 and 100: 11, 22, 33, 44, 55, 66, 77, 88, 99. Calculate this by dividing 100 by 11 (≈9.09) and taking the integer part.

Q14: What is the sum of the first 5 positive multiples of 11?

A14: The first 5 positive multiples of 11 are 11, 22, 33, 44, 55. Their sum is 11+22+33+44+55=165 (11×15=165).

Q15: Can the sum of two non-multiples of 11 be a multiple of 11?

A15: Yes. For example, 5 (not a multiple of 11) and 6 (not a multiple of 11) add up to 11 (a multiple of 11). Another example: 17 + 6 = 23 (no), but 18 + 4 = 22 (yes).

Q16: What is the 70th multiple of 11?

A16: The 70th multiple of 11 is calculated by multiplying 11 by 70: 11×70=770.

Q17: How do you teach multiples of 11 to young children?

A17: Use fun, hands-on methods: (1) Sing skip-counting songs for 11s; (2) Use manipulatives to group items into sets of 11; (3) Highlight 2-digit multiples (11,22…) with identical digits; (4) Connect to soccer teams (11 players) for real-world context.

Q18: Is 11 a multiple of itself?

A18: Yes. Every integer is a multiple of itself. For 11, this is shown by 11×1=11, so 11 is the first positive multiple of itself.

Q19: What is the least common multiple (LCM) of 11 and 13?

A19: The LCM of 11 and 13 is 143. Since both are prime numbers with no common factors, their LCM is their product (11×13=143).

Q20: Are multiples of 11 used in algebra?

A20: Yes. In algebra, multiples of 11 are used for factoring expressions (e.g., 11x+33=11(x+3)), solving linear equations (e.g., 11x=121 → x=11), and simplifying polynomial terms with coefficients that are multiples of 11.

Q21: How do you check if a very large number (e.g., 12,345,678) is a multiple of 11?

A21: Apply the divisibility rule:

  • Positions (right to left): 8(1),7(2),6(3),5(4),4(5),3(6),2(7),1(8)
  • Sum of odd positions: 8 + 6 + 4 + 2 = 20
  • Sum of even positions: 7 + 5 + 3 + 1 = 16
  • Difference: 20−16=4 (not 0 or multiple of 11) → Not a multiple of 11.

Q22: What is the relationship between multiples of 11 and 33?

A22: 33 is a multiple of 11 (11×3=33), and all multiples of 33 are multiples of 11 (e.g., 66, 99). Not all multiples of 11 are multiples of 33 (e.g., 11, 22).

Q23: Can a decimal number be a multiple of 11?

A23: No. By mathematical definition, multiples are strictly integers. Decimal numbers like 11.0 or 22.0 are not considered true multiples of 11—only whole numbers qualify.

Q24: Is the difference between two multiples of 11 always a multiple of 11?

A24: Yes. Let the two multiples be 11a and 11b (a,b are integers). Their difference is 11a−11b=11(a−b), which is a multiple of 11 because a−b is an integer.

Q25: What is the greatest common multiple of 11 and 44?

A25: There is no greatest common multiple. Multiples of any number are infinite, so they have no upper limit. Instead, find the greatest common factor (GCF = 11) or least common multiple (LCM = 44).

Q26: How do multiples of 11 relate to geometry?

A26: Multiples of 11 appear in geometric measurements for niche applications: (1) 11-sided polygons (hendecagons) use multiples of 11 for angle calculations; (2) Construction projects may use 11-foot beams for specialized structures; (3) Coordinate grids may use 11-unit increments for precision mapping.

Q27: Are multiples of 11 used in coding and programming?

A27: Yes. Programmers use multiples of 11 for: (1) Generating number pattern algorithms; (2) Setting intervals for automated tasks (e.g., running a check every 11 minutes); (3) Creating unique identifiers for data entries (e.g., 11, 22, 33 as category codes).

Q28: What is the smallest positive multiple of 11?

A28: The smallest positive multiple of 11 is 11 (when n=1). The smallest non-negative multiple of 11 is 0 (when n=0).

Q29: How do you find all multiples of 11 in a specific range (e.g., 500–600)?

A29: (1) Divide the lower bound by 11: 500÷11≈45.45 → next integer is 46; (2) 11×46=506 (first multiple ≥500); (3) Divide upper bound by 11: 600÷11≈54.54 → integer is 54; (4) 11×54=594 (last multiple ≤600); (5) List the sequence: 506, 517, 528, 539, 550, 561, 572, 583, 594.

Q30: Is 1,001 a multiple of 11?

A30: Yes. Sum of odd positions (1 + 0 = 1), sum of even positions (0 + 1 = 1), difference 0 → 1001 is a multiple of 11 (11×91=1001).

Q31: Can multiples of 11 be prime numbers?

A31: Only the number 11 itself is a prime multiple of 11. All other multiples of 11 (e.g., 22, 33) are composite numbers with at least three factors (1, 11, and the number itself).

Q32: How do multiples of 11 help with simplifying fractions?

A32: If both the numerator and denominator of a fraction are multiples of 11, divide both by 11 to simplify. Example: 12166​=121÷1166÷11​=116​; another example: 176143​=1613​.

Q33: What is the sum of all multiples of 11 from 1 to 200?

A33: The multiples are 11,22,…,198 (18 terms total). Use the arithmetic sequence sum formula: Sum=2n​×(first+last)=218​×(11+198)=9×209=1881.

Q34: Are multiples of 11 used in music and audio production?

A34: Yes, in niche ways: (1) 11-beat measures for experimental music; (2) 11 kHz audio sampling rates for low-bandwidth recordings; (3) 11-note musical scales in some traditional music styles.

Q35: How do you remember multiples of 11 easily?

A35: Use these tricks: (1) Memorize 2-digit multiples (identical digits); (2) For 3-digit multiples, remember the middle digit is the sum of first and last; (3) Practice skip counting daily to build fluency.

Q36: What is the digital root of a multiple of 11?

A36: The digital root (repeated digit sum) of multiples of 11 has no fixed pattern. Examples: 11 (1+1=2), 22 (4), 33 (6), 44 (8), 55 (10→1), 66 (12→3), etc.

Q37: Do multiples of 11 follow any unique patterns in the multiplication table?

A37: Yes. In the 11 times table: (1) 2-digit products have identical digits; (2) 3-digit products have a middle digit equal to the sum of the first and last digits; (3) Products alternate between odd and even.

Q38: Is 484 a multiple of 11?

A38: Yes. 11×44=484. Using the divisibility rule: sum of odd positions (4 + 4 = 8), sum of even positions (8), difference 0 → 484 is a multiple of 11.

Q39: What is the relationship between multiples of 11 and perfect squares?

A39: Some perfect squares are multiples of 11 (e.g., 112=121, 222=484), which are squares of multiples of 11. Most perfect squares (e.g., 16, 25, 36) are not multiples of 11, and most multiples of 11 are not perfect squares.

Q40: Can multiples of 11 be used to solve word problems efficiently?

A40: Yes. For example, if a soccer team has 11 players, 5 teams have 11×5=55 players total. For division problems (e.g., splitting 132 cookies into 11 bags), the answer is 12 cookies per bag.

Q41: What is the largest multiple of 11 less than 500?

A41: Divide 500 by 11 (≈45.45). Multiply 11 by 45: 11×45=495. So 495 is the largest multiple of 11 less than 500.

Q42: Are multiples of 11 used in game design?

A42: Yes. Game designers use multiples of 11 for: (1) Level milestones (unlocking rewards at level 11, 22); (2) Bonus point systems (110 points for a rare item); (3) Enemy spawn rates (spawning 11 enemies every wave); (4) Map dimensions (11×11 grids for puzzle levels).

Q43: How do multiples of 11 relate to modular arithmetic?

A43: In modular arithmetic, any multiple of 11 is congruent to 0 modulo 11 (11n≡0(mod11)). This property is used in check digit systems (e.g., ISBN numbers) to verify data accuracy.

Q44: Can the product of two non-multiples of 11 be a multiple of 11?

A44: No. Since 11 is prime, it can only divide a product if it divides at least one of the factors. If both factors are non-multiples of 11, their product will also be a non-multiple of 11.

Q45: What is the sum of the first 25 positive multiples of 11?

A45: The first 25 positive multiples form an arithmetic sequence: first term=11, last term=275 (11×25), n=25. Sum = 225​×(11+275)=12.5×286=3575.

Q46: Why are multiples of 11 important in number theory?

A46: In number theory, 11 is a prime number, so its multiples help illustrate properties of prime factorization, infinite sets, and divisibility rules for larger primes. They also appear in studies of palindromic numbers (e.g., 121, 1331).

Q47: Is 0 the only non-positive multiple of 11?

A47: No. Negative integers like -11, -22, -33 are also non-positive multiples of 11. 0 is the only multiple of 11 that is neither positive nor negative.

Q48: How do multiples of 11 help with estimating numbers?

A48: Rounding to the nearest multiple of 11 (e.g., 57 → 55, 70 → 66) simplifies mental math. This is useful for quick calculations in shopping, budgeting, or sports team planning.

Q49: What is the difference between multiples of 11 and multiples of 121?

A49: Multiples of 11 are numbers like 11, 22, 33 (divisible by 11), while multiples of 121 are numbers like 121, 242, 363 (divisible by 121 = 112). All multiples of 121 are multiples of 11, but not all multiples of 11 are multiples of 121.

Q50: Can multiples of 11 be used to teach prime factorization?

A50: Yes. Multiples of 11 (e.g., 22=2×11, 33=3×11, 44=2×2×11) are perfect for teaching prime factorization—they show how prime numbers combine to form composite numbers, with 11 as a core prime factor.

Q51: Is 1,331 a multiple of 11?

A51: Yes. 11×121=1331. Using the divisibility rule: sum of odd positions (1 + 3 = 4), sum of even positions (3 + 1 = 4), difference 0 → 1331 is a multiple of 11.

Q52: What is the 100th multiple of 11?

A52: The 100th multiple of 11 is 11×100=1100.

Q53: Are multiples of 11 used in cryptography?

A53: Yes, in some encryption algorithms. Prime numbers like 11 are used to generate keys, and multiples of 11 help with modular arithmetic operations that secure data transmission.

Q54: Can the sum of three non-multiples of 11 be a multiple of 11?

A54: Yes. For example, 1 + 2 + 8 = 11 (all non-multiples, sum is a multiple of 11). Another example: 5 + 6 + 0 = 11.

Q55: What is the smallest multiple of 11 that is also a multiple of 15?

A55: The smallest common multiple of 11 and 15 is 165. Since 11 and 15 are coprime, their LCM is their product (11×15=165).

8. Conclusion

Multiples of 11 are a fascinating subset of integers with unique digit patterns and properties, rooted in the fact that 11 is a prime number. Their distinct divisibility rules make them easy to identify, even for large numbers, and they appear in real-world contexts from sports teams to coding algorithms.

Whether you’re a student mastering divisibility rules, a teacher designing math lessons, a programmer building number pattern tools, or a gamer optimizing scoring systems, understanding multiples of 11 enhances your numerical literacy and problem-solving skills. From the simple 2-digit multiples with identical digits to the complex patterns of large numbers, multiples of 11 offer a window into the elegant structure of mathematics.