Divisibility Rules and Number Properties

TL;DR

Divisibility rules are handy shortcuts to check if one number divides another without long division. They rely on understanding basic number properties like factors, multiples, prime, and composite numbers. Mastering these rules helps you simplify fractions and solve number problems much faster.

1. The Mental Model

Think of divisibility rules as secret codes for numbers. Instead of doing a big calculation, these codes tell you instantly if a number can be split evenly into smaller groups. It's like knowing a secret handshake that lets you into the "divisible by" club!

2. The Core Material

Divisibility rules help you quickly determine if a number can be divided by another number without leaving a remainder. Let's look at the most common ones.

Divisibility Rules Cheat Sheet

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• By 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, 8).
  • Example: 346 is divisible by 2 because 6 is even.
• By 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: For 123, 1+2+3 = 6. Since 6 is divisible by 3, 123 is divisible by 3.
• By 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
  • Example: For 516, the last two digits form 16. Since 16 is divisible by 4, 516 is divisible by 4.
• By 5: A number is divisible by 5 if its last digit is 0 or 5.
  • Example: 780 is divisible by 5 because it ends in 0.
• By 6: A number is divisible by 6 if it is divisible by both 2 AND 3.
  • Example: For 42, it ends in 2 (divisible by 2) and 4+2=6 (divisible by 3). So, 42 is divisible by 6.
• By 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example: For 729, 7+2+9 = 18. Since 18 is divisible by 9, 729 is divisible by 9.
• By 10: A number is divisible by 10 if its last digit is 0.
  • Example: 150 is divisible by 10 because it ends in 0.

Number Properties Overview

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• Factors: Numbers that divide evenly into another number.
  • Example: The factors of 12 are 1, 2, 3, 4, 6, 12.
• Multiples: Numbers you get when you multiply a number by an integer.
  • Example: The multiples of 3 are 3, 6, 9, 12, ...
• Prime Numbers: A whole number greater than 1 that has only two factors: 1 and itself.
  • Example: 2, 3, 5, 7, 11 are prime numbers.
• Composite Numbers: A whole number greater than 1 that has more than two factors.
  • Example: 4, 6, 8, 9, 10 are composite numbers. (4 has factors 1, 2, 4).

How Divisibility Rules Help You

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These rules are super useful for:
1. Simplifying fractions: Quickly finding common factors.
2. Factoring numbers: Breaking down larger numbers into their prime factors.
3. Solving problems: Like finding the Least Common Multiple (LCM) or Greatest Common Divisor (GCD).

Here's how you might decide which rule to use:

graph TD
    Start["Given a Number (N)"] --> Check2["Is N divisible by 2? (ends in 0, 2, 4, 6, 8)"]
    Check2 -- Yes --> Is2["N is divisible by 2"]
    Check2 -- No --> Check5["Is N divisible by 5? (ends in 0, 5)"]
    Check5 -- Yes --> Is5["N is divisible by 5"]
    Check5 -- No --> Check10["Is N divisible by 10? (ends in 0)"]
    Check10 -- Yes --> Is10["N is divisible by 10"]
    Check10 -- No --> Check3["Is N divisible by 3? (sum of digits by 3)"]
    Check3 -- Yes --> Is3["N is divisible by 3"]
    Check3 -- No --> Check9["Is N divisible by 9? (sum of digits by 9)"]
    Check9 -- Yes --> Is9["N is divisible by 9"]
    Check9 -- No --> Check4["Is N divisible by 4? (last 2 digits by 4)"]
    Check4 -- Yes --> Is4["N is divisible by 4"]
    Check4 -- No --> End["No obvious rule, try division or other rules (e.g., 6)"]

    Is2 --> If6["If by 2 & 3, then by 6"]
    Is3 --> If6
    Is6["N is divisible by 6"]

    If6 --> Is6

3. Worked Example

Let's test the number 756. Is it divisible by 2, 3, 4, 5, 6, 9, or 10?

1. Divisible by 2? Yes, because 756 ends in 6 (an even number).
2. Divisible by 3? Sum of digits = 7 + 5 + 6 = 18. Yes, because 18 is divisible by 3 (18 ÷ 3 = 6).
3. Divisible by 4? Look at the last two digits: 56. Yes, because 56 is divisible by 4 (56 ÷ 4 = 14).
4. Divisible by 5? No, because 756 does not end in 0 or 5.
5. Divisible by 6? Yes, because it's divisible by both 2 AND 3.
6. Divisible by 9? Sum of digits = 18. Yes, because 18 is divisible by 9 (18 ÷ 9 = 2).
7. Divisible by 10? No, because 756 does not end in 0.

So, 756 is divisible by 2, 3, 4, 6, and 9.

4. Key Takeaways

• Divisibility rules are quick tests to see if a number can be divided evenly.
• Knowing divisibility by 2, 3, and 5 covers many common scenarios.
• The rule for 6 is a combination of the rules for 2 and 3.
• The rules for 3 and 9 both involve summing the digits of the number.
• Prime numbers have only two factors: 1 and themselves.

Common mistakes to avoid:
- Confusing the rule for 3 with the rule for 9; they are similar but different.
- Assuming a number divisible by 2 and 3 is also divisible by 5 (this is incorrect).
- Forgetting that the rule for 4 only cares about the last two digits, not all of them.
- Thinking 1 is a prime number; prime numbers must be greater than 1.

5. Now Try It

Take the number 2,340. Use the divisibility rules to determine if it is divisible by 2, 3, 4, 5, 6, 9, and 10. Write down your reasoning for each.

Success looks like: A list of "Yes" or "No" for each divisor (2, 3, 4, 5, 6, 9, 10), with a brief explanation matching the rules for each answer.