24 is a
refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
24 is a
twin-prime sum, specifically the sum of the third pair of twin primes .
24 is a
highly totient number, as there are 10 solutions to the equation
φ(x) = 24, which is more than any integer below 24.
144 (the
square of 12) and
576 (the square of 24) are also highly totient.[5]
24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24.
24 is the number of digits of the fifth and largest known
unitary perfect number, when written in
decimal: 146361946186458562560000.[7]
Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a
prime number; 24 is the largest number with this property.
24 is the largest
integer that is divisible by all
natural numbers no larger than its square root.
The product of any four consecutive numbers is divisible by 24. This is because, among any four consecutive numbers, there must be two even numbers, one of which is a multiple of four, and there must be at least one multiple of three.
24 = 4!, the
factorial of
4. It is the largest factorial that does not contain a trailing zero at the end of its digits (since factorial of any integer greater than 4 is divisible by both 2 and 5), and represents the number of ways to order 4 distinct items:
24 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 12, 24 — are exactly those numbers n for which every invertible element of the
commutative ringZ/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = {±1, ±5, ±7, ±11} is
isomorphic to the additive group (Z/2Z)3. This fact plays a role in
monstrous moonshine.
It follows that any number n relatively prime to 24 (that is, any number of the form 6K ± 1), and in particular any prime n greater than 3, has the property that n2 – 1 is divisible by 24.
24 is the Euler characteristic of a
K3 surface: a general elliptic K3 surface has exactly 24 singular fibers.
24 is the order of the
octahedral group — the group of rotations of the regular octahedron and the group of rotations of the cube. The
binary octahedral group is a subgroup of the 3-sphere S3 consisting of the 24 elements {±1, ±i, ±j, ±k, (±1±i±j±k)/2} of the binary tetrahedral group along with the 24 elements contained in its coset {(±1±i)/√2, (±1±j)/√2, (±1±k)/√2, (±i±j)/√2, (±i±k)/√2, (±j±k)/√2}. These two cosets each form the vertices of a self-dual
24-cell, and the two 24-cells are dual to each other. (See point below on 24-cell).
The vertices of the 24-cell honeycomb can be chosen so that in 4-dimensional space, identified with the ring of
quaternions, they are precisely the elements of the subring (the ring of "
Hurwitz integral quaternions") generated by the
binary tetrahedral group as represented by the set of 24 quaternions in the
D4 lattice. This set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere S3 of radius one centered at the origin. S3 is the Lie group Sp(1) of unit quaternions (isomorphic to the Lie groups SU(2) and Spin(3)), and so the binary tetrahedral group — of order 24 — is a subgroup of S3.
The 24 vertices of the 24-cell are contained in the
regular complex polygon4{3}4, or of symmetry order 1152, as well as 24 4-edges of 24 octahedral cells (of 48). Its representation in the
F4 Coxeter plane contains two rings of 12 vertices each.[9]
24 is the
kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a
24-cell).
In Christian
apocalyptic literature it represents the complete Church, being the sum of the 12
tribes of Israel and the 12
Apostles of the Lamb of God. For example, in The Book of Revelation: "Surrounding the throne were twenty-four other thrones, and seated on them were twenty-four elders. They were dressed in white and had crowns of gold on their heads."[11]
The number of
bits a computer needs to represent
24-bit color images (for a maximum of 16,777,216 colours—but greater numbers of bits provide more accurate colors).
The number of years from the start of the
Cold War until the signing of the
Seabed Arms Control Treaty, which banned the placing of nuclear weapons on the ocean floor within certain coastal distances.
The number of frames per second at which motion picture film is usually projected, as this is sufficient to allow for
persistence of vision.
The number of letters in both the modern and classical
Greek alphabet.[13] For the latter reason, also the number of chapters or "books" into which
Homer's Odyssey and Iliad came to be divided.
24 is a
refactorable number, as it has exactly eight positive divisors, and 8 is one of them.
24 is a
twin-prime sum, specifically the sum of the third pair of twin primes .
24 is a
highly totient number, as there are 10 solutions to the equation
φ(x) = 24, which is more than any integer below 24.
144 (the
square of 12) and
576 (the square of 24) are also highly totient.[5]
24's digits in decimal can be manipulated to form two of its factors, as 2 * 4 is 8 and 2 + 4 is 6. In turn 6 * 8 is 48, which is twice 24, and 4 + 8 is 12, which is half 24.
24 is the number of digits of the fifth and largest known
unitary perfect number, when written in
decimal: 146361946186458562560000.[7]
Subtracting 1 from any of its divisors (except 1 and 2 but including itself) yields a
prime number; 24 is the largest number with this property.
24 is the largest
integer that is divisible by all
natural numbers no larger than its square root.
The product of any four consecutive numbers is divisible by 24. This is because, among any four consecutive numbers, there must be two even numbers, one of which is a multiple of four, and there must be at least one multiple of three.
24 = 4!, the
factorial of
4. It is the largest factorial that does not contain a trailing zero at the end of its digits (since factorial of any integer greater than 4 is divisible by both 2 and 5), and represents the number of ways to order 4 distinct items:
24 is the only number whose divisors — 1, 2, 3, 4, 6, 8, 12, 24 — are exactly those numbers n for which every invertible element of the
commutative ringZ/nZ is a square root of 1. It follows that the multiplicative group of invertible elements (Z/24Z)× = {±1, ±5, ±7, ±11} is
isomorphic to the additive group (Z/2Z)3. This fact plays a role in
monstrous moonshine.
It follows that any number n relatively prime to 24 (that is, any number of the form 6K ± 1), and in particular any prime n greater than 3, has the property that n2 – 1 is divisible by 24.
24 is the Euler characteristic of a
K3 surface: a general elliptic K3 surface has exactly 24 singular fibers.
24 is the order of the
octahedral group — the group of rotations of the regular octahedron and the group of rotations of the cube. The
binary octahedral group is a subgroup of the 3-sphere S3 consisting of the 24 elements {±1, ±i, ±j, ±k, (±1±i±j±k)/2} of the binary tetrahedral group along with the 24 elements contained in its coset {(±1±i)/√2, (±1±j)/√2, (±1±k)/√2, (±i±j)/√2, (±i±k)/√2, (±j±k)/√2}. These two cosets each form the vertices of a self-dual
24-cell, and the two 24-cells are dual to each other. (See point below on 24-cell).
The vertices of the 24-cell honeycomb can be chosen so that in 4-dimensional space, identified with the ring of
quaternions, they are precisely the elements of the subring (the ring of "
Hurwitz integral quaternions") generated by the
binary tetrahedral group as represented by the set of 24 quaternions in the
D4 lattice. This set of 24 quaternions forms the set of vertices of a single 24-cell, all lying on the sphere S3 of radius one centered at the origin. S3 is the Lie group Sp(1) of unit quaternions (isomorphic to the Lie groups SU(2) and Spin(3)), and so the binary tetrahedral group — of order 24 — is a subgroup of S3.
The 24 vertices of the 24-cell are contained in the
regular complex polygon4{3}4, or of symmetry order 1152, as well as 24 4-edges of 24 octahedral cells (of 48). Its representation in the
F4 Coxeter plane contains two rings of 12 vertices each.[9]
24 is the
kissing number in 4-dimensional space: the maximum number of unit spheres that can all touch another unit sphere without overlapping. (The centers of 24 such spheres form the vertices of a
24-cell).
In Christian
apocalyptic literature it represents the complete Church, being the sum of the 12
tribes of Israel and the 12
Apostles of the Lamb of God. For example, in The Book of Revelation: "Surrounding the throne were twenty-four other thrones, and seated on them were twenty-four elders. They were dressed in white and had crowns of gold on their heads."[11]
The number of
bits a computer needs to represent
24-bit color images (for a maximum of 16,777,216 colours—but greater numbers of bits provide more accurate colors).
The number of years from the start of the
Cold War until the signing of the
Seabed Arms Control Treaty, which banned the placing of nuclear weapons on the ocean floor within certain coastal distances.
The number of frames per second at which motion picture film is usually projected, as this is sufficient to allow for
persistence of vision.
The number of letters in both the modern and classical
Greek alphabet.[13] For the latter reason, also the number of chapters or "books" into which
Homer's Odyssey and Iliad came to be divided.