Divisibility by 9
A number is divisible by 9 if its digit sum is.
Formula
Variables
Example
729: 7+2+9=18 ✓
How it works
A number is divisible by 9 when the sum of its digits is divisible by 9. This works because each place value is a power of 10, and powers of 10 leave a remainder of 1 when divided by 9, so the whole number has the same remainder as its digit sum. In other words, replacing each digit with its value and adding them gives a quick test for divisibility by 9.
When to use it
Use this when you want a fast mental check for whether a number can be divided by 9 with no remainder. It is especially useful in arithmetic, factor checks, and spotting errors in calculations.
Worked example, step by step
Check whether 5,418 is divisible by 9. Step 1: Add the digits: 5 + 4 + 1 + 8 = 18 Step 2: Check the digit sum: 18 is divisible by 9 Step 3: Therefore, 5,418 is divisible by 9 Final answer: Yes, 5,418 is divisible by 9.
Common mistakes to avoid
- Using subtraction instead of adding the digits
- Stopping after one digit sum without checking whether that sum is divisible by 9
- Confusing the rule for 9 with the rule for 3
Did You Know?
The digits of any multiple of 9 always sum to a multiple of 9, the basis of the old "casting out nines" error check.
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