Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is.
Formula
Variables
Example
123: 1+2+3=6 ✓
How it works
The reason this works is that 10 leaves a remainder of 1 when divided by 3, so every power of 10 also acts like 1 modulo 3. That means a number written in base 10 has the same remainder mod 3 as the sum of its digits. So if the digit sum is a multiple of 3, the whole number is too.
When to use it
Use this rule to quickly test whether a number is divisible by 3 without long division. It is especially helpful in mental math, checking factors, and simplifying fractions or algebraic expressions.
Worked example, step by step
Check whether 582 is divisible by 3. 5 + 8 + 2 = 15 15 is divisible by 3, so 582 is divisible by 3. Therefore, 582 ÷ 3 = 194.
Common mistakes to avoid
- Adding the digits incorrectly before testing divisibility
- Forgetting that the rule tests divisibility by 3, not by 6 or 9
- Stopping after one digit sum without checking whether the sum itself is divisible by 3
Did You Know?
A number is divisible by 3 exactly when its digits sum to a multiple of 3 — try it on 12,345.
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