Number dictionary · Exact arithmetic
513: factors and number facts
five hundred thirteen is an odd number and is composite. Its 8 positive divisors form 4 factor pairs.
Number JD Editorial Team · Updated 2026-10-08
- In words
- five hundred thirteen
- Roman numeral
- DXIII
- Prime factorization
- 3^3 × 19
- Digital root
- 9
All factors of 513
A positive factor divides 513 with no remainder. In ascending order, the complete list is:
1, 3, 9, 19, 27, 57, 171, 513
Factor pairs
- 1 × 513 = 513
- 3 × 171 = 513
- 9 × 57 = 513
- 19 × 27 = 513
Each pair describes an arrangement of 513 items into equal rows. The pairs have different row and column counts; 513 cannot make a square array of whole items.
Divisor totals
- Positive divisors
- 8
- Sum of all divisors
- 800
- Sum excluding 513
- 287
- Classification
- deficient
513 is deficient because its proper divisors add to 287, which is less than 513.
Prime factorization of 513
513 = 3^3 × 19
Repeated division, step by step
- 513 ÷ 3 = 171
- 171 ÷ 3 = 57
- 57 ÷ 3 = 19
- 19 ÷ 19 = 1
Multiply the prime divisors in these steps to reconstruct 513. An exponent records how many times the same prime occurs. The divisor-count calculation is (3 + 1) × (1 + 1) = 8.
The preceding prime is 509. The next larger prime is 521. The factor list above explains why 513 is composite.
Learn the factorization method513 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | five hundred thirteen |
| Roman numeral | DXIII |
| Binary (base 2) | 1000000001 |
| Octal (base 8) | 1001 |
| Decimal (base 10) | 513 |
| Hexadecimal (base 16) | 201 |
Check the binary conversion
Read 1000000001 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
512 + 1 = 513
Check the hexadecimal conversion
2 × 16^2 + 0 × 16^1 + 1 × 16^0 = 513
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 513
- 513 squared
- 513 × 513 = 263,169
- 513 cubed
- 513 × 513 × 513 = 135,005,697
- Square root
- ≈ 22.649503
- Simplified radical
- 3√57
513 lies between 484 (22²) and 529 (23²), so it is not a perfect square. It is not a perfect cube.
513 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 513.
Distinguish a square from a square rootMultiplication table for 513
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 | 256 | 1 | No |
| 3 | 171 | 0 | Yes |
| 4 | 128 | 1 | No |
| 5 | 102 | 3 | No |
| 6 | 85 | 3 | No |
| 7 | 73 | 2 | No |
| 8 | 64 | 1 | No |
| 9 | 57 | 0 | Yes |
| 10 | 51 | 3 | No |
| 11 | 46 | 7 | No |
| 12 | 42 | 9 | No |
For example, 513 = 7 × 73 + 2. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 513
513 contains 3 decimal digits. Its digit sum is 5 + 1 + 3 = 9.
- 513: 5 + 1 + 3 = 9
The digital root is 9. Reversing its digits gives 315; it is not 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, 513/100 simplifies to 513/100 by dividing both terms by their greatest common divisor, 1. Its decimal value is 5.13. Because the numerator exceeds the denominator, this value is greater than one.
The greatest common divisor of 513 and 12 is 3. Their least common multiple is 2052. The identity GCD × LCM = 513 × 12 = 6156 checks both results.
Work through GCF and LCMQuestions about 513
What are the factors of 513?
The positive factors are 1, 3, 9, 19, 27, 57, 171, 513. There are 8 factors, including 1 and 513.
Is 513 a prime number?
No. 513 is composite: 3^3 × 19 = 513.
How do you write 513 in Roman numerals?
513 is DXIII in standard subtractive Roman numeral notation.
What is the binary representation of 513?
513 in decimal is 1000000001 in base 2. Adding its active place values gives 512 + 1 = 513.
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.
NUMBER JOURNAL / JD
A reflective note on 513
Meaning varies by tradition and personal experience. One sighting is not evidence of what will happen next.
The digits of 513 add to 9. That arithmetic result is separate from the cultural symbolism of the pattern. If this number matters to you, write down what you noticed and choose one grounded action rather than treating the sighting as a prediction.