Basic Divisibility Rules

Divisibility rules are handy shortcuts that let you quickly figure out if a number can be divided evenly by another number—without having to do the actual long division!

Rules Based on the Last Digit (2, 5, and 10)

For some numbers, you only need to look at the very last digit to know if it is divisible.

• Divisible by 2: A number is divisible by 2 if its last digit is an even number ($0, 2, 4, 6,$ or $8$).
  • Example: $34\textbf{6}$ is divisible by 2. $12\textbf{7}$ is not.
• Divisible by 5: A number is divisible by 5 if its last digit is $0$ or $5$.
  • Example: $89\textbf{5}$ and $1{,}00\textbf{0}$ are divisible by 5.
• Divisible by 10: A number is divisible by 10 if its last digit is $0$.
  • Example: $4{,}52\textbf{0}$ is divisible by 10.

Rules Based on the Sum of Digits (3 and 9)

For 3 and 9, you need to add up all the digits in the number.

• Divisible by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
  • Example: For $252$, add the digits: $2 + 5 + 2 = 9$. Since $9$ is divisible by 3, $252$ is divisible by 3.
• Divisible by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
  • Example: For $819$, add the digits: $8 + 1 + 9 = 18$. Since $18$ is divisible by 9, $819$ is divisible by 9.

Combining Rules (Divisible by 6)

Sometimes you can combine rules. To check if a number is divisible by 6, it must pass the rules for both 2 and 3.

• It must end in an even number.
• Its digits must add up to a multiple of 3.

Example Problems

1. Is $4{,}536$ divisible by both 2 and 3?

• Check 2: The last digit is $6$ (an even number), so yes, it is divisible by 2.
• Check 3: Add the digits: $4 + 5 + 3 + 6 = 18$. Since $18$ is divisible by 3, yes, it is divisible by 3.
• Bonus: Because it is divisible by both 2 and 3, $4{,}536$ is also divisible by 6!

2. Find all numbers between 1 and 100 that are divisible by both 3 and 5.

• To be divisible by 5, the numbers must end in $0$ or $5$.
• To be divisible by 3, the sum of their digits must be a multiple of 3.
• Let's check the multiples of 5 under 100:
  • $15$ ($1+5=6$, yes)
  • $30$ ($3+0=3$, yes)
  • $45$ ($4+5=9$, yes)
  • $60$ ($6+0=6$, yes)
  • $75$ ($7+5=12$, yes)
  • $90$ ($9+0=9$, yes)
• The numbers are: $15, 30, 45, 60, 75, 90$.

3. Is $78$ a multiple of 6?

• To be a multiple of 6, it must be divisible by both 2 and 3.
• Check 2: The last digit is $8$ (even), so yes.
• Check 3: Add the digits: $7 + 8 = 15$. Since $15$ is divisible by 3, yes.
• Since it passes both tests, yes, $78$ is a multiple of 6.