Number dictionary · Exact arithmetic
9: factors and number facts
nine is an odd number and is composite. Its 3 positive divisors form 2 factor pairs.
Number JD Editorial Team · Updated 2026-10-08
- In words
- nine
- Roman numeral
- IX
- Prime factorization
- 3^2
- Digital root
- 9
All factors of 9
A positive factor divides 9 with no remainder. In ascending order, the complete list is:
1, 3, 9
Factor pairs
- 1 × 9 = 9
- 3 × 3 = 9
Each pair describes an arrangement of 9 items into equal rows. The 3 × 3 pair is square, so its shared factor appears only once in the factor list.
Divisor totals
- Positive divisors
- 3
- Sum of all divisors
- 13
- Sum excluding 9
- 4
- Classification
- deficient
9 is deficient because its proper divisors add to 4, which is less than 9.
Prime factorization of 9
9 = 3^2
Repeated division, step by step
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
Multiply the prime divisors in these steps to reconstruct 9. An exponent records how many times the same prime occurs. The divisor-count calculation is (2 + 1) = 3.
The preceding prime is 7. The next larger prime is 11. The factor list above explains why 9 is composite.
Learn the factorization method9 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | nine |
| Roman numeral | IX |
| Binary (base 2) | 1001 |
| Octal (base 8) | 11 |
| Decimal (base 10) | 9 |
| Hexadecimal (base 16) | 9 |
Check the binary conversion
Read 1001 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
8 + 1 = 9
Check the hexadecimal conversion
9 × 16^0 = 9
Hexadecimal uses A through F for digit values 10 through 15. These notations represent the same quantity; changing the base does not change the integer.
Square, cube and square root of 9
- 9 squared
- 9 × 9 = 81
- 9 cubed
- 9 × 9 × 9 = 729
- Square root
- = 3
- Simplified radical
- 3
9 is a perfect square: 3 × 3 = 9. It is not a perfect cube.
9 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 9.
Distinguish a square from a square rootMultiplication table for 9
These are the first twelve positive multiples. Unlike the finite factor list, the sequence of multiples continues without an upper limit.
Divisibility checks
| Divide by | Whole quotient | Remainder | Divides exactly? |
|---|---|---|---|
| 2 | 4 | 1 | No |
| 3 | 3 | 0 | Yes |
| 4 | 2 | 1 | No |
| 5 | 1 | 4 | No |
| 6 | 1 | 3 | No |
| 7 | 1 | 2 | No |
| 8 | 1 | 1 | No |
| 9 | 1 | 0 | Yes |
| 10 | 0 | 9 | No |
| 11 | 0 | 9 | No |
| 12 | 0 | 9 | No |
For example, 9 = 7 × 1 + 2. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 9
9 contains 1 decimal digit. Its digit sum is 9 = 9.
It already has one digit, so there is no reduction step.
The digital root is 9. 9 reads the same forward and backward, making it a decimal palindrome. Digit patterns depend on the notation being used.
Digital-root arithmetic always finishes at a single digit for a positive integer. Some numerology methods stop at 11, 22 or 33 instead; that is a separate interpretive convention.
Fractions and a worked comparison
As a fraction of 100, 9/100 simplifies to 9/100 by dividing both terms by their greatest common divisor, 1. Its decimal value is 0.09.
The greatest common divisor of 9 and 12 is 3. Their least common multiple is 36. The identity GCD × LCM = 9 × 12 = 108 checks both results.
Work through GCF and LCMQuestions about 9
What are the factors of 9?
The positive factors are 1, 3, 9. There are 3 factors, including 1 and 9.
Is 9 a prime number?
No. 9 is composite: 3^2 = 9.
How do you write 9 in Roman numerals?
9 is IX in standard subtractive Roman numeral notation.
What is the binary representation of 9?
9 in decimal is 1001 in base 2. Adding its active place values gives 8 + 1 = 9.
Methods and reference definitions
Number JD calculates each result from the integer on this page. Positive factors are enumerated in pairs; repeated division gives the prime factorization; place values verify base conversions. Decimal square roots are rounded to six places when irrational.