Divisibility by 2 (Even Numbers)

Is $246$ divisible by $2$?

Look at the last digit: The last digit of $246$ is $6$ = Last digit: $6$

Check if it's even (0, 2, 4, 6, or 8): $6$ is an even number = Yes, $6$ is even

Apply the rule: If the last digit is even, the whole number is divisible by $2$ = Divisible by $2$

Answer: Yes, $246$ is divisible by $2$ because it ends in $6$ (an even digit).

Divisibility by 3 (Sum of Digits)

Is $531$ divisible by $3$?

Add all the digits together: $5 + 3 + 1 = 9$ = Digit sum: $9$

Check if the sum is divisible by 3: $9 \div 3 = 3$ with no remainder = Yes, $9$ is divisible by $3$

Apply the rule: If the digit sum is divisible by $3$, the whole number is too = Divisible by $3$

Answer: Yes, $531$ is divisible by $3$ because $5 + 3 + 1 = 9$, and $9$ is divisible by $3$.

Divisibility by 4 (Last Two Digits)

Is $1,324$ divisible by $4$?

Look at the last two digits: The last two digits of $1,324$ are $24$ = Last two digits: $24$

Check if this number is divisible by 4: $24 \div 4 = 6$ with no remainder = Yes, $24$ is divisible by $4$

Apply the rule: If the last two digits form a number divisible by $4$, the whole number is too = Divisible by $4$

Answer: Yes, $1,324$ is divisible by $4$ because the last two digits ($24$) are divisible by $4$.

Divisibility by 5 (Last Digit)

Is $7,835$ divisible by $5$?

Look at the last digit: The last digit of $7,835$ is $5$ = Last digit: $5$

Check if it's 0 or 5: $5$ is one of the target digits = Yes, it's $5$

Apply the rule: If the last digit is $0$ or $5$, the number is divisible by $5$ = Divisible by $5$

Answer: Yes, $7,835$ is divisible by $5$ because it ends in $5$.

Divisibility by 6 (Combined Rules)

Is $432$ divisible by $6$?

Check divisibility by 2: Last digit is $2$ (even), so divisible by $2$ = Divisible by $2$: Yes

Check divisibility by 3: $4 + 3 + 2 = 9$, and $9 \div 3 = 3$ = Divisible by $3$: Yes

Apply the rule for 6: If divisible by BOTH $2$ AND $3$, then divisible by $6$ = Divisible by $6$

Answer: Yes, $432$ is divisible by $6$ because it's divisible by both $2$ (ends in $2$) and $3$ (digits sum to $9$).

Divisibility by 9 (Sum of Digits)

Is $2,457$ divisible by $9$?

Add all the digits together: $2 + 4 + 5 + 7 = 18$ = Digit sum: $18$

Check if the sum is divisible by 9: $18 \div 9 = 2$ with no remainder = Yes, $18$ is divisible by $9$

Apply the rule: If the digit sum is divisible by $9$, the whole number is too = Divisible by $9$

Answer: Yes, $2,457$ is divisible by $9$ because $2 + 4 + 5 + 7 = 18$, and $18$ is divisible by $9$.

Non-Example: When Rules Fail

Is $523$ divisible by $3$?

Add all the digits together: $5 + 2 + 3 = 10$ = Digit sum: $10$

Check if the sum is divisible by 3: $10 \div 3 = 3$ R $1$ = No, $10$ is NOT divisible by $3$

Apply the rule: Since the digit sum is not divisible by $3$, neither is the original = NOT divisible by $3$

Answer: No, $523$ is NOT divisible by $3$ because $5 + 2 + 3 = 10$, and $10$ is not divisible by $3$.

Mistake: Using the last digit rule for divisibility by 3

Why: Unlike divisibility by 2 or 5, the last digit alone doesn't tell you about divisibility by 3. For example, $13$ ends in $3$ but is NOT divisible by $3$.

Correct: For divisibility by 3, add ALL the digits together and check if that sum is divisible by 3.

Mistake: Checking only divisibility by 2 OR 3 for divisibility by 6

Why: A number must pass BOTH tests. For example, $9$ is divisible by $3$ but not by $2$, so it's not divisible by $6$.

Correct: For divisibility by 6, the number must be divisible by BOTH 2 AND 3.

Mistake: Confusing the rules for 3 and 9

Why: Both rules involve summing digits, but they check different divisors. A number divisible by 9 is always divisible by 3, but not vice versa.

Correct: For 3: digit sum divisible by 3. For 9: digit sum divisible by 9. Example: $12$ (digits sum to $3$) is divisible by $3$ but not by $9$.

Mistake: Looking at only the last digit for divisibility by 4

Why: Unlike 2 and 5, you need the last TWO digits for 4. For example, $14$ ends in $4$ but $14 \div 4 = 3$ R $2$.

Correct: For divisibility by 4, check if the last TWO digits form a number divisible by 4.

Splitting Costs Evenly

When splitting a bill among friends, divisibility rules help you know instantly if it will work out evenly.

A restaurant bill is 72 dollars. Can 6 people split it evenly? Check: $72$ ends in $2$ (divisible by $2$) and $7 + 2 = 9$ (divisible by $3$), so yes, divisible by $6$. Each person pays 12 dollars.

Organizing Items in Groups

Teachers, coaches, and event planners use divisibility to organize people or items into equal groups.

A teacher has 28 students. Can she make groups of 4? Check: Last two digits are $28$, and $28 \div 4 = 7$. Yes! She can make 7 groups of 4.

Q: Why does the sum of digits work for 3 and 9?

A: It's related to place value! Each place (ones, tens, hundreds) leaves a remainder of 1 when divided by 9 (or 3). So the remainder of the whole number equals the remainder of the digit sum.

Q: Is there a rule for divisibility by 7?

A: Yes, but it's more complicated! Double the last digit, subtract it from the rest, and check if the result is divisible by 7. For most cases, it's easier to just divide.

Q: If a number is divisible by 9, is it also divisible by 3?

A: Yes! Since 9 = 3 times 3, any multiple of 9 is automatically a multiple of 3. But not vice versa: 12 is divisible by 3 but not by 9.