Divisibility Rules in Depth

TL;DR

Divisibility rules are mental shortcuts to quickly check if one number divides another without doing long division. They work by exploiting properties of our base-10 number system. Mastering these rules helps you simplify fractions, factor numbers, and generally become more comfortable with number properties.

1. The Mental Model

Think of divisibility rules as secret codes for numbers. Instead of decoding the whole message (the full division), you just look for a specific pattern or "key" in the number to know if it's divisible by another.

2. The Core Material

Divisibility rules let us check if a number is perfectly divisible by another (meaning no remainder). These aren't magic; they're based on number theory and place value. Knowing them makes mental math much easier.

Divisibility by 2, 5, and 10 (Last Digit Rules)

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These are the easiest because they only look at the last digit.

• By 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8).
  • Example: 346 is divisible by 2 because 6 is even. 71 is not because 1 is odd.
• By 5: A number is divisible by 5 if its last digit is 0 or 5.
  • Example: 105 is divisible by 5 because it ends in 5. 238 is not.
• By 10: A number is divisible by 10 if its last digit is 0.
  • Example: 420 is divisible by 10 because it ends in 0. 55 is not.

Divisibility by 3 and 9 (Sum of Digits Rules)

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These rules require you to add up all the digits of the number.

• By 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: For 573: 5 + 7 + 3 = 15. Since 15 is divisible by 3 (15 ÷ 3 = 5), then 573 is divisible by 3.
• By 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example: For 189: 1 + 8 + 9 = 18. Since 18 is divisible by 9 (18 ÷ 9 = 2), then 189 is divisible by 9.
  • Note: If a number is divisible by 9, it's automatically divisible by 3. But not the other way around! (e.g., 12 is divisible by 3 but not 9).

Divisibility by 4 and 8 (Last Digit Group Rules)

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These rules look at groups of digits from the end of the number.

• By 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Example: For 1,236, look at 36. Since 36 is divisible by 4 (36 ÷ 4 = 9), then 1,236 is divisible by 4.
  • Shortcut: If the tens digit is even, and the last digit is 0, 4, or 8, it's divisible by 4. If the tens digit is odd, and the last digit is 2 or 6, it's divisible by 4.
• By 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
  • Example: For 7,128, look at 128. Since 128 is divisible by 8 (128 ÷ 8 = 16), then 7,128 is divisible by 8.
  • This rule is a bit harder for mental math, but it's consistent.

Divisibility by 6 (Combined Rule)

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• By 6: A number is divisible by 6 if it's divisible by both 2 and 3.
  • Example: For 456:
    1. Is it divisible by 2? Yes, because it ends in 6 (even).
    2. Is it divisible by 3? Sum of digits: 4 + 5 + 6 = 15. Yes, because 15 is divisible by 3.
       Since it passes both, 456 is divisible by 6.

Divisibility by 7 (A Trickier Rule)

The rule for 7 is a bit more involved and often it's quicker to just do the division for smaller numbers. However, here it is:

• By 7: Take the last digit, double it, and subtract it from the rest of the number. If the result is 0 or divisible by 7, then the original number is divisible by 7. Repeat if the number is still large.

  • Example: For 343:

    1. Last digit is 3. Double it: 3 * 2 = 6.
    2. Remaining number is 34. Subtract 6: 34 - 6 = 28.
    3. Since 28 is divisible by 7 (28 ÷ 7 = 4), then 343 is divisible by 7.

  • Example: For 1,603:

    1. Last digit is 3. Double it: 6.
    2. Remaining number is 160. Subtract 6: 160 - 6 = 154.
    3. Repeat for 154: Last digit is 4. Double it: 8.
    4. Remaining number is 15. Subtract 8: 15 - 8 = 7.
    5. Since 7 is divisible by 7, then 1,603 is divisible by 7.

Divisibility by 11 (Alternating Sum Rule)

• By 11: Sum the alternating digits. If the result is 0 or divisible by 11, the original number is divisible by 11.

  • Example: For 1,364:

    1. (Sum of odd-placed digits) - (Sum of even-placed digits)
    2. (4 + 3) - (6 + 1) = 7 - 7 = 0.
    3. Since the result is 0, 1,364 is divisible by 11.

  • Example: For 9,157:

    1. (7 + 1) - (5 + 9) = 8 - 14 = -6.
    2. Since -6 is not 0 or divisible by 11, 9,157 is not divisible by 11.

graph TD
    A["Number (N)"] --> B{Last Digit?};
    B --> C{Ends in 0, 2, 4, 6, 8?};
    C -- Yes --> D["Divisible by 2"];
    C -- No --> E["Not Divisible by 2"];
    B --> F{Ends in 0 or 5?};
    F -- Yes --> G["Divisible by 5"];
    F -- No --> H["Not Divisible by 5"];
    B --> I{Ends in 0?};
    I -- Yes --> J["Divisible by 10"];
    I -- No --> K["Not Divisible by 10"];

    A --> L{Sum of Digits?};
    L --> M{Sum % 3 == 0?};
    M -- Yes --> N["Divisible by 3"];
    M -- No --> O["Not Divisible by 3"];
    L --> P{Sum % 9 == 0?};
    P -- Yes --> Q["Divisible by 9"];
    P -- No --> R["Not Divisible by 9"];

    A --> S{Last 2 Digits?};
    S --> T{Last 2-digit number % 4 == 0?};
    T -- Yes --> U["Divisible by 4"];
    T -- No --> V["Not Divisible by 4"];

    A --> W{Last 3 Digits?};
    W --> X{Last 3-digit number % 8 == 0?};
    X -- Yes --> Y["Divisible by 8"];
    X -- No --> Z["Not Divisible by 8"];

    A --> AA{Is N Divisible by 2 AND 3?};
    AA -- Yes --> BB["Divisible by 6"];
    AA -- No --> CC["Not Divisible by 6"];

    A --> DD{Alternating Sum of Digits?};
    DD --> EE{Alternating Sum % 11 == 0?};
    EE -- Yes --> FF["Divisible by 11"];
    EE -- No --> GG["Not Divisible by 11"];

    A --> HH{Repeated Subtracting 2x Last Digit?};
    HH --> II{Result is 0 or % 7 == 0?};
    II -- Yes --> JJ["Divisible by 7"];
    II -- No --> KK["Not Divisible by 7"];

3. Worked Example

Let's test the number 7,392 for divisibility by 2, 3, 4, 5, 6, 8, 9, 10, and 11.

• By 2: Ends in 2 (even). Yes.
• By 3: Sum of digits: 7 + 3 + 9 + 2 = 21. Is 21 divisible by 3? Yes (21 ÷ 3 = 7). Yes.
• By 4: Last two digits form 92. Is 92 divisible by 4? Yes (92 ÷ 4 = 23). Yes.
• By 5: Doesn't end in 0 or 5 (ends in 2). No.
• By 6: Divisible by both 2 and 3? Yes, we found it is. Yes.
• By 8: Last three digits form 392. Is 392 divisible by 8? Yes (392 ÷ 8 = 49). Yes.
• By 9: Sum of digits is 21. Is 21 divisible by 9? No. No.
• By 10: Doesn't end in 0 (ends in 2