Skill: Apply divisibility rules

A number is divisible by another number if the division has no remainder. For example, if $(24 \div 6=4)$, then $(24)$ is divisible by $6$, and $6$ is a factor of $(24)$.

Use these divisibility rules:

• 2: The last digit is even: $(0,2,4,6,)$ or $8$.
• 3: The sum of the digits is divisible by $3$.
• 4: The last two digits form a number divisible by $4$.
• 5: The last digit is $0$ or $5$.
• 6: The number is divisible by both $2$ and $3$.
• 8: The last three digits form a number divisible by $8$.
• 9: The sum of the digits is divisible by $9$.
• 10: The last digit is $0$.

Worked example

Determine which of $(2,3,4,5,6,8,9,)$ and $(10)$ divide $(3{,}240)$.

1. Test for $2$:
   The last digit is $0$, which is even.
   So, $(3{,}240)$ is divisible by $2$.

2. Test for $3$:
   Add the digits:

$$3+2+4+0=9$$

Since $9$ is divisible by $3$, $(3{,}240)$ is divisible by $3$.

3. Test for $4$:
   Look at the last two digits: $(40)$.
   Since $(40\div4=10)$, $(3{,}240)$ is divisible by $4$.

4. Test for $5$:
   The last digit is $0$, so $(3{,}240)$ is divisible by $5$.

5. Test for $6$:
   The number is divisible by both $2$ and $3$, so it is divisible by $6$.

6. Test for $8$:
   Look at the last three digits: $(240)$.
   Since $(240\div8=30)$, $(3{,}240)$ is divisible by $8$.

7. Test for $9$:
   The digit sum is $9$, and $9$ is divisible by $9$.
   Therefore, $(3{,}240)$ is divisible by $9$.

8. Test for $(10)$:
   The last digit is $0$, so $(3{,}240)$ is divisible by $(10)$.

Therefore, $(3{,}240)$ is divisible by all eight numbers:

$$2,\ 3,\ 4,\ 5,\ 6,\ 8,\ 9,\text{ and }10.$$

A number can satisfy several divisibility tests at the same time.