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Modulo 9 Checker

Calculate the sum of a number's digits, find its remainder modulo 9, check whether it is divisible by 9, and determine its digital root using the mathematical digit-sum rule.

Modulo 9 Divisibility by 9 Digit Sum Digital Root Very Large Integers Free Math Tool

Check a Number Modulo 9

Enter any non-negative integer. The calculator works with the number as text, so it can also process integers much larger than the normal PHP integer range.

Use digits only. Negative numbers and decimals are not accepted.
Try: 9, 12345, 999999

The Divisibility Rule for 9

A whole number is divisible by 9 if and only if the sum of its decimal digits is divisible by 9.

\[ n \equiv \text{sum of digits of }n \pmod{9} \]

This means that you do not need to perform ordinary long division to test divisibility by 9. You can simply add the digits and inspect the resulting remainder.

Why does this work?

In base 10, every power of 10 is congruent to 1 modulo 9:

\[ 10 \equiv 1 \pmod{9} \] \[ 10^2 \equiv 1 \pmod{9} \] \[ 10^3 \equiv 1 \pmod{9} \]

Consider a decimal number such as 5724:

\[ 5724 = 5(10^3)+7(10^2)+2(10)+4 \]
\[ 5724 \equiv 5+7+2+4 = 18 \pmod{9} \]
\[ 18 \equiv 0 \pmod{9} \]

Therefore 5724 is divisible by 9.

Modulo 9 Examples

Example 1: 123

1 + 2 + 3 = 6
6 mod 9 = 6
Not divisible by 9

Example 2: 729

7 + 2 + 9 = 18
18 mod 9 = 0
Divisible by 9

Example 3: 1008

1 + 0 + 0 + 8 = 9
9 mod 9 = 0
Divisible by 9

Example 4: 123456789

1+2+3+4+5+6+7+8+9 = 45
45 mod 9 = 0
Divisible by 9

Modulo 9 and Digital Roots

Modulo 9 is closely connected to the digital root of an integer. Repeatedly adding the digits of a positive number until a single digit remains produces its digital root.

\[ \operatorname{digital\ root}(n) = \begin{cases} 0, & n=0 \\ 9, & n\equiv0\pmod9,\ n>0 \\ n\bmod9, & \text{otherwise} \end{cases} \]

Example

Take the number 987654:

\[ 9+8+7+6+5+4=39 \]
\[ 3+9=12 \]
\[ 1+2=3 \]

Therefore the digital root is 3, and the original number has remainder 3 modulo 9.

For a dedicated tool, see the Digital Root Calculator .

Applications of Modulo 9

The digit-sum property of modulo 9 appears in elementary number theory, mental arithmetic, divisibility tests and error-detection techniques.

1. Quick divisibility testing

You can test divisibility by 9 without calculating the complete quotient. If the digit sum is a multiple of 9, the original number is also a multiple of 9.

2. Checking arithmetic

A modulo-9 check can sometimes be used as a quick consistency test for arithmetic calculations. It is not a complete proof that a calculation is correct, because different numbers can have the same remainder modulo 9.

3. Number theory

Congruences modulo 9 are useful when studying divisibility, remainders, digit patterns and properties of integers. For larger problems, modulo arithmetic can reduce complicated expressions to much smaller values.

4. Digital roots

Repeated digit summation is effectively a convenient way to calculate a number's residue modulo 9, with the special case that a positive multiple of 9 has digital root 9.

Modulo 9 Compared with Other Divisibility Rules

Divisibility by 3

A number is divisible by 3 when its digit sum is divisible by 3.

Divisibility by 9

A number is divisible by 9 when its digit sum is divisible by 9.

Divisibility by 2

A decimal integer is divisible by 2 when its final digit is even.

Divisibility by 5

A decimal integer is divisible by 5 when its final digit is 0 or 5.

For more number-theory tools, explore the Prime Factorization Calculator , GCD Calculator , and LCM Calculator .

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Modulo 9 FAQ

What does modulo 9 mean?

Modulo 9 means finding the remainder after dividing an integer by 9. The possible remainders are 0 through 8.

How do I check if a number is divisible by 9?

Add all of its decimal digits. If the resulting digit sum is divisible by 9, the original number is divisible by 9.

Why does the digit-sum rule work for 9?

Because 10 is congruent to 1 modulo 9. Consequently every power of 10 is also congruent to 1 modulo 9, allowing a decimal number to be replaced by the sum of its digits without changing its remainder modulo 9.

What is the difference between modulo 9 and digital root?

Modulo 9 returns a remainder from 0 through 8. The digital root of a positive integer returns 1 through 9, with positive multiples of 9 having digital root 9. Zero has digital root 0.

Can this calculator handle very large numbers?

Yes. The calculator processes the input as a string rather than converting the entire value into a PHP integer. This allows very long non-negative decimal integers to be checked without normal integer overflow.

What is the remainder if the digit sum is 18?

Since 18 is divisible by 9, its remainder modulo 9 is 0. Therefore the original number is also divisible by 9.

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