Number dictionary · Exact arithmetic
31: factors and number facts
thirty-one is an odd number and is prime. Its 2 positive divisors form 1 factor pair.
Number JD Editorial Team · Updated 2026-10-08
- In words
- thirty-one
- Roman numeral
- XXXI
- Prime factorization
- 31
- Digital root
- 4
All factors of 31
A positive factor divides 31 with no remainder. In ascending order, the complete list is:
1, 31
Factor pairs
- 1 × 31 = 31
Each pair describes an arrangement of 31 items into equal rows. The pairs have different row and column counts; 31 cannot make a square array of whole items.
Divisor totals
- Positive divisors
- 2
- Sum of all divisors
- 32
- Sum excluding 31
- 1
- Classification
- deficient
31 is deficient because its proper divisors add to 1, which is less than 31.
Prime factorization of 31
31 = 31
Repeated division, step by step
- 31 ÷ 31 = 1
Multiply the prime divisors in these steps to reconstruct 31. Each prime in this factorization occurs once. The divisor-count calculation is (1 + 1) = 2.
The preceding prime is 29. The next larger prime is 37. To check 31 for primality, it is sufficient to test possible divisors from 2 through 5.
Learn the factorization method31 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | thirty-one |
| Roman numeral | XXXI |
| Binary (base 2) | 11111 |
| Octal (base 8) | 37 |
| Decimal (base 10) | 31 |
| Hexadecimal (base 16) | 1F |
Check the binary conversion
Read 11111 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
16 + 8 + 4 + 2 + 1 = 31
Check the hexadecimal conversion
1 × 16^1 + 15 × 16^0 = 31
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 31
- 31 squared
- 31 × 31 = 961
- 31 cubed
- 31 × 31 × 31 = 29,791
- Square root
- ≈ 5.567764
- Simplified radical
- √31
31 lies between 25 (5²) and 36 (6²), so it is not a perfect square. It is not a perfect cube.
31 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 31.
Distinguish a square from a square rootMultiplication table for 31
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 | 15 | 1 | No |
| 3 | 10 | 1 | No |
| 4 | 7 | 3 | No |
| 5 | 6 | 1 | No |
| 6 | 5 | 1 | No |
| 7 | 4 | 3 | No |
| 8 | 3 | 7 | No |
| 9 | 3 | 4 | No |
| 10 | 3 | 1 | No |
| 11 | 2 | 9 | No |
| 12 | 2 | 7 | No |
For example, 31 = 7 × 4 + 3. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 31
31 contains 2 decimal digits. Its digit sum is 3 + 1 = 4.
- 31: 3 + 1 = 4
The digital root is 4. Reversing its digits gives 13; 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, 31/100 simplifies to 31/100 by dividing both terms by their greatest common divisor, 1. Its decimal value is 0.31.
The greatest common divisor of 31 and 12 is 1. Their least common multiple is 372. The identity GCD × LCM = 31 × 12 = 372 checks both results.
Work through GCF and LCMQuestions about 31
What are the factors of 31?
The positive factors are 1, 31. There are 2 factors, including 1 and 31.
Is 31 a prime number?
Yes. The only positive divisors of 31 are 1 and 31.
How do you write 31 in Roman numerals?
31 is XXXI in standard subtractive Roman numeral notation.
What is the binary representation of 31?
31 in decimal is 11111 in base 2. Adding its active place values gives 16 + 8 + 4 + 2 + 1 = 31.
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 31
Meaning varies by tradition and personal experience. One sighting is not evidence of what will happen next.
The digits of 31 add to 4. 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.