Number dictionary · Exact arithmetic
821: factors and number facts
eight hundred twenty-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
- eight hundred twenty-one
- Roman numeral
- DCCCXXI
- Prime factorization
- 821
- Digital root
- 2
All factors of 821
A positive factor divides 821 with no remainder. In ascending order, the complete list is:
1, 821
Factor pairs
- 1 × 821 = 821
Each pair describes an arrangement of 821 items into equal rows. The pairs have different row and column counts; 821 cannot make a square array of whole items.
Divisor totals
- Positive divisors
- 2
- Sum of all divisors
- 822
- Sum excluding 821
- 1
- Classification
- deficient
821 is deficient because its proper divisors add to 1, which is less than 821.
Prime factorization of 821
821 = 821
Repeated division, step by step
- 821 ÷ 821 = 1
Multiply the prime divisors in these steps to reconstruct 821. Each prime in this factorization occurs once. The divisor-count calculation is (1 + 1) = 2.
The preceding prime is 811. The next larger prime is 823. To check 821 for primality, it is sufficient to test possible divisors from 2 through 28.
Learn the factorization method821 in words, Roman numerals and other bases
| Representation | Value |
|---|---|
| English (US style) | eight hundred twenty-one |
| Roman numeral | DCCCXXI |
| Binary (base 2) | 1100110101 |
| Octal (base 8) | 1465 |
| Decimal (base 10) | 821 |
| Hexadecimal (base 16) | 335 |
Check the binary conversion
Read 1100110101 from right to left using place values 1, 2, 4, 8 and successive powers of two. Keep the values whose bit is 1:
512 + 256 + 32 + 16 + 4 + 1 = 821
Check the hexadecimal conversion
3 × 16^2 + 3 × 16^1 + 5 × 16^0 = 821
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 821
- 821 squared
- 821 × 821 = 674,041
- 821 cubed
- 821 × 821 × 821 = 553,387,661
- Square root
- ≈ 28.653098
- Simplified radical
- √821
821 lies between 784 (28²) and 841 (29²), so it is not a perfect square. It is not a perfect cube.
821 is not triangular: no positive whole-number k satisfies k(k + 1) / 2 = 821.
Distinguish a square from a square rootMultiplication table for 821
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 | 410 | 1 | No |
| 3 | 273 | 2 | No |
| 4 | 205 | 1 | No |
| 5 | 164 | 1 | No |
| 6 | 136 | 5 | No |
| 7 | 117 | 2 | No |
| 8 | 102 | 5 | No |
| 9 | 91 | 2 | No |
| 10 | 82 | 1 | No |
| 11 | 74 | 7 | No |
| 12 | 68 | 5 | No |
For example, 821 = 7 × 117 + 2. A remainder of zero is what makes a divisor a factor.
Digit sum and digital root of 821
821 contains 3 decimal digits. Its digit sum is 8 + 2 + 1 = 11.
- 821: 8 + 2 + 1 = 11
- 11: 1 + 1 = 2
The digital root is 2. Reversing its digits gives 128; 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, 821/100 simplifies to 821/100 by dividing both terms by their greatest common divisor, 1. Its decimal value is 8.21. Because the numerator exceeds the denominator, this value is greater than one.
The greatest common divisor of 821 and 12 is 1. Their least common multiple is 9852. The identity GCD × LCM = 821 × 12 = 9852 checks both results.
Work through GCF and LCMQuestions about 821
What are the factors of 821?
The positive factors are 1, 821. There are 2 factors, including 1 and 821.
Is 821 a prime number?
Yes. The only positive divisors of 821 are 1 and 821.
How do you write 821 in Roman numerals?
821 is DCCCXXI in standard subtractive Roman numeral notation.
What is the binary representation of 821?
821 in decimal is 1100110101 in base 2. Adding its active place values gives 512 + 256 + 32 + 16 + 4 + 1 = 821.
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 821
Meaning varies by tradition and personal experience. One sighting is not evidence of what will happen next.
The digits of 821 add to 11. 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.