How to Solve: Divisibility Rules

How to Solve: Divisibility Rules

Introduction

"Imagine you’re in an exam, and you have 30 seconds to decide if 4,825 is divisible by 5—without doing long division. Mastering divisibility rules lets you answer instantly, saving time for harder questions. Let’s break it down."

What You Need To Know First

1. Place value (units, tens, hundreds, etc.).
2. Basic multiplication facts (e.g., multiples of 2, 5, 10).
3. Sum of digits (e.g., the sum of digits in 123 is 1 + 2 + 3 = 6).

Key Vocabulary

Formulas To Know

(All must be memorized—no exam sheet will provide these!)

1. Divisible by 2?
2. Rule: The last digit is even (0, 2, 4, 6, 8).

3. Example: 346 → last digit is 6 (even) → divisible by 2.

4. Divisible by 3?

5. Rule: Sum of digits is divisible by 3.

6. Example: 123 → 1 + 2 + 3 = 6 → 6 ÷ 3 = 2 → divisible by 3.

7. Divisible by 4?

8. Rule: The last two digits form a number divisible by 4.

9. Example: 1,324 → last two digits = 24 → 24 ÷ 4 = 6 → divisible by 4.

10. Divisible by 5?

11. Rule: The last digit is 0 or 5.

12. Example: 2,340 → last digit is 0 → divisible by 5.

13. Divisible by 6?

14. Rule: Must be divisible by both 2 and 3.

15. Example: 132 → last digit is 2 (divisible by 2) → sum of digits = 6 (divisible by 3) → divisible by 6.

16. Divisible by 9?

17. Rule: Sum of digits is divisible by 9.

18. Example: 729 → 7 + 2 + 9 = 18 → 18 ÷ 9 = 2 → divisible by 9.

19. Divisible by 10?

20. Rule: The last digit is 0.
21. Example: 5,670 → last digit is 0 → divisible by 10.

Step-by-Step Method

How to Check Divisibility (Any Number)

1. Identify the divisor (e.g., 2, 3, 4, 5, 6, 9, 10).
2. Apply the correct rule from the list above.
3. Check the condition (last digit, sum of digits, etc.).
4. If the condition is met → divisible. If not → not divisible.
5. For 6: Must pass both the 2 and 3 rules.

Worked Example (Using Steps): Is 1,236 divisible by 6? 1. Divisor = 6 → must check rules for 2 and 3. 2. Rule for 2: Last digit = 6 (even) → ✅ passes. 3. Rule for 3: Sum of digits = 1 + 2 + 3 + 6 = 12 → 12 ÷ 3 = 4 → ✅ passes. 4. Both rules passed → 1,236 is divisible by 6.

Worked Examples

Example 1 - Basic

Is 45 divisible by 5? 1. Divisor = 5 → check last digit. 2. Last digit = 5 → ✅ passes. 3. Answer: Yes, 45 is divisible by 5.

What we did and why: - We used the 5 rule (last digit 0 or 5). - No calculation needed—just look at the last digit.

Example 2 - Medium

Is 1,044 divisible by 4? 1. Divisor = 4 → check last two digits. 2. Last two digits = 44 → 44 ÷ 4 = 11 → ✅ passes. 3. Answer: Yes, 1,044 is divisible by 4.

What we did and why: - The 4 rule requires checking the last two digits, not just the last one. - We divided 44 by 4 to confirm.

Example 3 - Exam Style

Which of these numbers is divisible by 9?
A) 2,345
B) 3,618
C) 4,520
D) 5,671

1. Divisor = 9 → check sum of digits.
2. A) 2,345 → 2 + 3 + 4 + 5 = 14 → 14 ÷ 9 = 1 R5 → ❌ fails.
3. B) 3,618 → 3 + 6 + 1 + 8 = 18 → 18 ÷ 9 = 2 → ✅ passes.
4. C) 4,520 → 4 + 5 + 2 + 0 = 11 → ❌ fails.
5. D) 5,671 → 5 + 6 + 7 + 1 = 19 → ❌ fails.
6. Answer: B) 3,618.

What we did and why: - We applied the 9 rule to each option. - Only B had a digit sum divisible by 9.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Divisibility rules are your secret weapon for quick answers. Here’s the cheat sheet: - 2? Last digit even. - 3? Sum of digits ÷ 3. - 4? Last two digits ÷ 4. - 5? Last digit 0 or 5. - 6? Passes both 2 and 3. - 9? Sum of digits ÷ 9. - 10? Last digit 0.

Before the exam, grab a random number and test each rule. If you mess up, check the sum or last digits again. You’ve got this—now go ace those questions!