Divisibility Rules: 1 to 13 Guide
Quickly check if numbers divide evenly with these easy-to-learn rules!
Have you ever wondered if there’s a quick way to know if 546 is divisible by 3 without doing the actual division? There is! Divisibility rules are your secret weapon for mental math, helping you check if one number can be divided by another with no remainder.
What is Divisibility? 🤔
A number is divisible by another number if you can divide them and get a whole number answer with a remainder of zero. For example, 12 is divisible by 3 because $12 \div 3 = 4$, which is a whole number.
Why Learn Divisibility Rules?
- ⚡ Speed: Check for factors in seconds.
- 📝 Simplify Fractions: Easily find common factors to reduce fractions.
- 🧩 Problem Solving: They are essential for topics like Prime Factorisation, HCF, and LCM.
🎯 Easy Rules (1, 2, 3, 5, 10)
These are the most common and easiest rules to remember. Let’s start with them!
Rule for 1: Every integer is divisible by 1. (Easy, right?)
Rule for 2: A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).
Example: Is 3,498 divisible by 2?
Yes, because it ends in 8, which is an even number.
Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Example: Is 546 divisible by 3?
Let’s add the digits: $5 + 4 + 6 = 15$. Since 15 is divisible by 3 ($15 \div 3 = 5$), the number 546 is divisible by 3.
Rule for 5: A number is divisible by 5 if its last digit is a 0 or a 5.
Example: Is 12,375 divisible by 5?
Yes, because it ends in 5.
Rule for 10: A number is divisible by 10 if its last digit is 0.
Example: Is 9,990 divisible by 10?
Yes, because it ends in 0.
🧩 Intermediate Rules (4, 6, 8, 9, 12)
These rules often combine other rules or look at more than just the last digit.
Rule for 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
Example: Is 7,324 divisible by 4?
Look at the last two digits: 24. Since 24 is divisible by 4 ($24 \div 4 = 6$), the number 7,324 is divisible by 4.
Rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Example: Is 822 divisible by 6?
1. Check for 2: It ends in 2, so it’s divisible by 2. ✅
2. Check for 3: The sum of its digits is $8 + 2 + 2 = 12$. Since 12 is divisible by 3, the number is divisible by 3. ✅
Because it passes both tests, 822 is divisible by 6.
Rule for 8: A number is divisible by 8 if the number formed by its last three digits is divisible by 8.
Example: Is 51,120 divisible by 8?
Look at the last three digits: 120. Let’s check if 120 is divisible by 8. $120 \div 8 = 15$. Yes, it is. Therefore, 51,120 is divisible by 8.
Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9. (Similar to the rule for 3!)
Example: Is 2,853 divisible by 9?
Let’s add the digits: $2 + 8 + 5 + 3 = 18$. Since 18 is divisible by 9 ($18 \div 9 = 2$), the number 2,853 is divisible by 9.
Rule for 12: A number is divisible by 12 if it is divisible by both 3 and 4.
Example: Is 636 divisible by 12?
1. Check for 3: The sum of digits is $6 + 3 + 6 = 15$. 15 is divisible by 3. ✅
2. Check for 4: The last two digits are 36. 36 is divisible by 4. ✅
Because it passes both tests, 636 is divisible by 12.
⚡ Advanced Rules (7, 11, 13)
These rules are a bit more complex but are fantastic tools for your math kit. They often require a small calculation.
Rule for 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, the original number is divisible by 7. You can repeat the process if the number is still large.
Example: Is 343 divisible by 7?
1. The last digit is 3. The rest of the number is 34.
2. Double the last digit: $3 \times 2 = 6$.
3. Subtract this from the rest: $34 – 6 = 28$.
4. Is 28 divisible by 7? Yes, $28 \div 7 = 4$. So, 343 is divisible by 7.
Rule for 11: Find the alternating sum of the digits (first digit – second digit + third digit – …). If the result is 0 or divisible by 11, the original number is divisible by 11.
Example: Is 5,841 divisible by 11?
1. Alternating sum: $5 – 8 + 4 – 1$.
2. Calculate: $(5+4) – (8+1) = 9 – 9 = 0$.
3. The result is 0, so 5,841 is divisible by 11.
Rule for 13: Take the last digit, multiply it by 4, and add it to the rest of the number. If the result is divisible by 13, the original number is divisible by 13. You can repeat the process.
Example: Is 273 divisible by 13?
1. The last digit is 3. The rest of the number is 27.
2. Multiply the last digit by 4: $3 \times 4 = 12$.
3. Add this to the rest: $27 + 12 = 39$.
4. Is 39 divisible by 13? Yes, $39 \div 13 = 3$. So, 273 is divisible by 13.
Summary of Divisibility Rules
- 1: All numbers.
- 2: Last digit is even.
- 3: Sum of digits is divisible by 3.
- 4: Last two digits are divisible by 4.
- 5: Last digit is 0 or 5.
- 6: Divisible by 2 AND 3.
- 7: Double last digit, subtract from rest. Result is divisible by 7.
- 8: Last three digits are divisible by 8.
- 9: Sum of digits is divisible by 9.
- 10: Last digit is 0.
- 11: Alternating sum of digits is 0 or divisible by 11.
- 12: Divisible by 3 AND 4.
- 13: Multiply last digit by 4, add to rest. Result is divisible by 13.
Practice Problems
Time to test your skills! Use the divisibility rules to solve these problems. Show your reasoning.
1. Is the number 4,152 divisible by 6?
View Answer
To be divisible by 6, a number must be divisible by both 2 and 3.
Test for 2: The number 4,152 ends in 2, which is an even digit. So, it is divisible by 2.
Test for 3: Sum the digits: $4 + 1 + 5 + 2 = 12$. Since 12 is divisible by 3, the number is divisible by 3.
Conclusion: Because 4,152 is divisible by both 2 and 3, it is divisible by 6.
2. Is the number 81,927 divisible by 9?
View Answer
To be divisible by 9, the sum of the digits must be divisible by 9.
Sum the digits: $8 + 1 + 9 + 2 + 7 = 27$.
Is 27 divisible by 9? Yes, $27 div 9 = 3$.
Conclusion: Yes, 81,927 is divisible by 9.
3. Is the number 1,495 divisible by 7?
View Answer
We use the rule for 7: double the last digit and subtract it from the rest of the number.
The last digit is 5. The rest of the number is 149.
Double the last digit: $5 times 2 = 10$.
Subtract from the rest: $149 - 10 = 139$.
Is 139 divisible by 7? It's not obvious, so let's apply the rule again to 139.
The last digit is 9. The rest is 13. Double the last digit: $9 times 2 = 18$. Subtract: $13 - 18 = -5$.
Since -5 is not 0 or divisible by 7, the number 1,495 is not divisible by 7.
4. Which of the following numbers is divisible by 12: 3,424 or 5,136?
View Answer
A number is divisible by 12 if it is divisible by both 3 and 4.
Let's test 3,424:
Test for 3: Sum of digits is $3+4+2+4 = 13$. 13 is not divisible by 3. So, 3,424 is not divisible by 12.
Let's test 5,136:
Test for 3: Sum of digits is $5+1+3+6 = 15$. 15 is divisible by 3. ✅
Test for 4: The last two digits are 36. 36 is divisible by 4 ($36 div 4 = 9$). ✅
Conclusion: Because 5,136 is divisible by both 3 and 4, it is divisible by 12.
Interactive Quiz
Check your understanding with this quick quiz!