Divisibility rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
- Identify the last digit of the number
- Check if the last digit is even (0, 2, 4, 6, 8)
- If even, the number is divisible by 2
- If odd, the number is not divisible by 2
The number is 3,456
The last digit is 6
Even digits: 0, 2, 4, 6, 8
Since 6 is in this list, it is even
Since the last digit is even, the number is divisible by 2
3,456 ÷ 2 = 1,728
Since there's no remainder, the number is indeed divisible by 2
Yes, 3,456 is divisible by 2 because its last digit (6) is even.
• Rule for 2: Last digit must be even
• Even digits: 0, 2, 4, 6, 8
• Odd digits: 1, 3, 5, 7, 9
Divisibility rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
Digits of 4,827: 4, 8, 2, 7
Sum = 4 + 8 + 2 + 7 = 21
Is 21 divisible by 3?
21 ÷ 3 = 7 with no remainder
Since 21 is divisible by 3, the original number 4,827 is also divisible by 3
4,827 ÷ 3 = 1,609
Since there's no remainder, the number is divisible by 3
Yes, 4,827 is divisible by 3 because the sum of its digits (21) is divisible by 3.
• Rule for 3: Sum of digits must be divisible by 3
• Digit addition: Add each digit individually
• Verification: Division confirms the rule
Divisibility rule for 5: A number is divisible by 5 if its last digit is 0 or 5.
1,230: Last digit is 0
4,567: Last digit is 7
8,905: Last digit is 5
3,421: Last digit is 1
Rule: Last digit must be 0 or 5
1,230: 0 → Divisible by 5
4,567: 7 → Not divisible by 5
8,905: 5 → Divisible by 5
3,421: 1 → Not divisible by 5
Numbers divisible by 5: 1,230 and 8,905
1,230 ÷ 5 = 246 ✓
8,905 ÷ 5 = 1,781 ✓
Numbers divisible by 5 are 1,230 and 8,905 because their last digits are 0 and 5 respectively.
• Rule for 5: Last digit must be 0 or 5
• Binary check: Only two acceptable endings
• Quick identification: Just examine the last digit
Divisibility rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
Digits of 7,389: 7, 3, 8, 9
Sum = 7 + 3 + 8 + 9 = 27
Is 27 divisible by 9?
27 ÷ 9 = 3 with no remainder
Since 27 is divisible by 9, the original number 7,389 is also divisible by 9
7,389 ÷ 9 = 821
Since there's no remainder, the number is divisible by 9
Yes, 7,389 is divisible by 9 because the sum of its digits (27) is divisible by 9.
• Rule for 9: Sum of digits must be divisible by 9
• Similar to 3: Same digit sum principle
• Verification: Division confirms the rule
Divisibility rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3.
Last digit of 4,632 is 2
Since 2 is even, the number is divisible by 2
Sum of digits: 4 + 6 + 3 + 2 = 15
Is 15 divisible by 3? 15 ÷ 3 = 5 with no remainder
Yes, 15 is divisible by 3
Since 4,632 is divisible by both 2 and 3, it is divisible by 6
4,632 ÷ 6 = 772
Since there's no remainder, the number is divisible by 6
Yes, 4,632 is divisible by 6 because it is divisible by both 2 (last digit is even) and 3 (digit sum is 15, which is divisible by 3).
• Rule for 6: Must satisfy both rules for 2 and 3
• Compound rule: Both conditions must be true
• Verification: Check both conditions independently
Divisibility Rules Reference
Divisibility: A number a is divisible by another number b if a ÷ b results in a whole number with no remainder.
Factor: A number that divides another number evenly without leaving a remainder.
Multiple: The product of a number and an integer (e.g., multiples of 3: 3, 6, 9, 12, 15...).
Prime number: A number greater than 1 that has only two factors: 1 and itself.
Composite number: A number greater than 1 that has more than two factors.
Remainder: The amount left over when one number does not divide evenly into another.
- Identify the divisor: Determine which number you're testing divisibility by
- Select the rule: Choose the appropriate divisibility rule for that number
- Apply the rule: Follow the specific steps of the chosen rule
- Calculate: Perform the necessary arithmetic
- Check result: Determine if the condition is met
- Verify: Optionally divide to confirm the result
• Rule for 2: Last digit is even (0, 2, 4, 6, 8)
• Rule for 3: Sum of digits is divisible by 3
• Rule for 4: Last two digits form a number divisible by 4
• Rule for 5: Last digit is 0 or 5
• Rule for 6: Divisible by both 2 and 3
• Rule for 9: Sum of digits is divisible by 9
• Rule for 10: Last digit is 0
Rule for 4: Check if the last two digits form a number divisible by 4
Last two digits: 24
Is 24 divisible by 4? 24 ÷ 4 = 6 ✓
Yes, 5,724 is divisible by 4
Rule for 10: Last digit must be 0
3,450: Last digit is 0 → Divisible by 10
7,892: Last digit is 2 → Not divisible by 10
10,005: Last digit is 5 → Not divisible by 10
5,600: Last digit is 0 → Divisible by 10
Numbers divisible by 10: 3,450 and 5,600
Rule for 12: Must be divisible by both 3 and 4
Check for 3: 4+8+9+6 = 27, 27÷3 = 9 ✓
Check for 4: Last two digits are 96, 96÷4 = 24 ✓
Since divisible by both 3 and 4, 4,896 is divisible by 12
Number: 2,520
2: Last digit is 0 (even) ✓
3: 2+5+2+0 = 9, 9÷3 = 3 ✓
4: Last two digits 20, 20÷4 = 5 ✓
5: Last digit is 0 ✓
6: Divisible by 2 and 3 ✓
7: 2520÷7 = 360 ✓
8: Last three digits 520, 520÷8 = 65 ✓
9: 2+5+2+0 = 9, 9÷9 = 1 ✓
10: Last digit is 0 ✓
Divisible by: 2, 3, 4, 5, 6, 7, 8, 9, 10
84: Divisible by 2 (even), so 84 = 2 × 42
42: Divisible by 2 (even), so 42 = 2 × 21
21: Divisible by 3 (2+1=3), so 21 = 3 × 7
7: Prime number
Prime factorization: 84 = 2² × 3 × 7
Questions & Answers
Question: Why do we need divisibility rules? Can't I just divide the number to see if it goes in evenly?
Answer: Great question! Divisibility rules are valuable shortcuts that save time and mental energy:
- Speed: Checking the last digit is faster than long division
- Mental math: You can quickly determine divisibility without paper
- Large numbers: Much easier to check divisibility of large numbers
- Problem solving: Useful for factoring, simplifying fractions, finding multiples
For example, to check if 1,234,567 is divisible by 3, just add: 1+2+3+4+5+6+7 = 28. Since 28 isn't divisible by 3, neither is the large number!
Question: My child is getting confused with the divisibility rules. How can I help them remember them better?
Answer: Here are memory aids for divisibility rules:
- Rule 2: "Even Steven" - even numbers are divisible by 2
- Rule 3: "Add me up" - add digits and see if sum is divisible by 3
- Rule 5: "Five or Zero" - ends in 5 or 0
- Rule 6: "Both Two and Three" - must satisfy both rules
- Rule 9: "Add to Nine" - like rule 3 but for 9
- Rule 10: "Zero Hero" - must end in 0
Practice with different numbers regularly and use flashcards to reinforce the rules!
Question: What if a number is divisible by more than one number? Like 12 - is it divisible by 2, 3, 4, and 6?
Answer: Yes, absolutely! Many numbers are divisible by multiple numbers:
- 12 is divisible by: 1, 2, 3, 4, 6, 12
- These are called factors of 12
- Every number is divisible by 1 and itself
- Some numbers like 12 have many factors (called composite numbers)
- Prime numbers only have 2 factors: 1 and themselves
For 12: 12÷1=12, 12÷2=6, 12÷3=4, 12÷4=3, 12÷6=2, 12÷12=1
So yes, 12 satisfies the divisibility rules for 2, 3, 4, and 6!
Divisibility: A number is divisible by another number if the division results in a whole number with no remainder.
Divisibility Rule: A shortcut method to determine if one number divides another evenly without performing the actual division.
Factor: A number that divides another number evenly without leaving a remainder.
Multiple: The result of multiplying a number by an integer (e.g., multiples of 4: 4, 8, 12, 16...).
Prime Number: A number greater than 1 that has exactly two factors: 1 and itself.
Composite Number: A number greater than 1 that has more than two factors.
Remainder: The amount left over when one number does not divide evenly into another.
- Identify the divisor: Determine which number you're testing divisibility by (2, 3, 4, 5, 6, etc.).
- Select the appropriate rule: Choose the specific divisibility rule for that number.
- Apply the rule: Follow the exact steps of the chosen rule methodically.
- Perform calculations: Carry out any necessary arithmetic (adding digits, checking last digits).
- Check conditions: Determine if the conditions of the rule are met.
- State the result: Clearly indicate whether the number is divisible or not.
- Verify if needed: Optionally perform the actual division to confirm your result.
• Rule for 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8)
• Rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3
• Rule for 4: A number is divisible by 4 if the last two digits form a number divisible by 4
• Rule for 5: A number is divisible by 5 if its last digit is 0 or 5
• Rule for 6: A number is divisible by 6 if it is divisible by both 2 and 3
• Rule for 9: A number is divisible by 9 if the sum of its digits is divisible by 9
• Rule for 10: A number is divisible by 10 if its last digit is 0