561
561 is the smallest Carmichael number — a composite that passes Fermat's primality test for every base.
- Odd
- Composite
Where 561 fits
Why 561 is special
Carmichael numbers are why fast primality tests must be probabilistic and repeated rather than trusted once. 561 factors as 3 x 11 x 17 and still behaves like a prime under the test, which is precisely the failure mode cryptographic software has to defend against.
561 is part of
Factors & digit properties
- Divisors (8)
- 1, 3, 11, 17, 33, 51, 187, 561
- Prime factorisation
- 3 × 11 × 17
- Digit sum
- 12
- Digital root
- 3
Representations
- Decimal
- 561
- Scientific notation
- 5.610 × 10^2
Roman numerals
D L X I
Binary
1 0 0 0 1 1 0 0 0 1
1 × 512 + 0 × 256 + 0 × 128 + 0 × 64 + 1 × 32 + 1 × 16 + 0 × 8 + 0 × 4 + 0 × 2 + 1 × 1 = 561
Octal
1 0 6 1
1 × 512 + 0 × 64 + 6 × 8 + 1 × 1 = 561
Hexadecimal
2 3 1
2 × 256 + 3 × 16 + 1 × 1 = 561
Read as powers of two: 512 + 32 + 16 + 1 = 561.
In historical numeral systems
Egyptian
5 coils of rope, 6 hobbles, 1 stroke
5 × 100 + 6 × 10 + 1 × 1 = 561. The signs are simply repeated and added; where they sit on the line is a matter of layout, not value.
Babylonian
9 vertical wedges (sixties place), 2 corner wedges + 1 vertical wedge (units place)
9 × 60 + 21 = 561. Places are separated by a space on the tablet, with the higher place written first.
Greek (Ionic)
phi, then xi, then alpha, marked as a numeral
phi-xi-alpha + keraia
phi + xi + alpha written in descending order and added — the letters carry their values wherever they stand, so this is addition, not place value.
Hebrew numerals
tav, then qof, then samekh, then alef
tav-qof-samekh-alef
tav (400) + qof (100) + samekh (60) + alef (1) = 561, written in descending order.
Systems that simply cannot reach this value are left out. Where a script needs a font most readers do not have, the reading is given instead of the glyphs — see number systems for each system in full.
Other mathematical patterns
- Triangular number
- Deficient number
- Hexagonal number
Technology & computing
Because composites can pass Fermat's test, cryptographic libraries use the stronger Miller–Rabin test and run it repeatedly with random bases, each round cutting the chance of a false positive by at least three quarters. Carmichael numbers are the concrete reason: they defeat the naive test completely rather than occasionally. Alford, Granville and Pomerance proved in 1994 that infinitely many exist, so the problem cannot be sidestepped by working above some threshold.
DocumentedTechnology & computingPeriod: infinitude proved 1994
Mathematics
Alwin Korselt characterised these numbers in 1899: n is a Carmichael number exactly when it is squarefree and p − 1 divides n − 1 for every prime p dividing n. 561 satisfies it — 2, 10 and 16 all divide 560. Korselt did not identify a single number meeting his own criterion; Robert Carmichael found 561 in 1910, eleven years later, which is why the criterion and the numbers carry different names.
DocumentedMathematicsPeriod: criterion 1899, first example 1910561 is the smallest Carmichael number: it factors as 3 x 11 x 17 and still satisfies Fermat's primality test for every base coprime to it. Such numbers are why practical primality testing is probabilistic and repeated rather than trusted once, which matters directly to key generation.
DocumentedMathematics
Common questions
What is a Carmichael number?
A composite that passes Fermat's primality test for every base coprime to it — so the test cannot tell it from a prime. 561 is the smallest, factoring as 3 × 11 × 17. They are why real primality testing uses Miller–Rabin with repeated random bases rather than trusting one result, and why the problem cannot be sidestepped by working above some threshold: infinitely many exist, proved in 1994.
Is 561 a prime number?
No, 561 is not prime.
What are the factors of 561?
The factors of 561 are 1, 3, 11, 17, 33, 51, 187, 561.
What is 561 in Roman numerals?
561 in Roman numerals is DLXI.
About this page's data
Mathematical facts:computed directly from the number's value — nothing to source or verify externally.
Contextual records: 3 reviewer-confirmed records, sourced and reviewed before publication.
Last reviewed: 15 July 2026