# 216 (number)

 ← 215 216 217 →
Cardinal two hundred sixteen
Ordinal 216th
(two hundred and sixteenth)
Factorization 23× 33
Divisors 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
Roman numeral CCXVI
Binary 110110002
Ternary 220003
Quaternary 31204
Quinary 13315
Senary 10006
Octal 3308
Duodecimal 16012
Vigesimal AG20
Base 36 6036

216 (two hundred [and] sixteen) is the natural number following 215 and preceding 217.

Since $216 = 3^3 + 4^3 + 5^3 = 6^3$, it is the smallest cube that is also the sum of three cubes (Plato was among the first to notice this, and mentioned it in Book VIII of Republic). It is also the sum of a twin prime (107 + 109). But since there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number. This multiplicative magic square

$\begin{pmatrix} 2 & 9 & 12\\ 36 & 6 & 1\\ 3 & 4 & 18 \end{pmatrix}$

has magic constant 216.

216 is the number of digits in the decimal system of the matrix formed by the Fibonacci sequence in module $9$ for each natural number in module $9$.[1]

$1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9$

$2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9, 7, 7, 5, 3, 8, 2, 1, 3, 4, 7, 2, 9$

$3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9$

$4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5, 9, 5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9$

$5, 5, 1, 6, 7, 4, 2, 6, 8, 5, 4, 9, 4, 4, 8, 3, 2, 5, 7, 3, 1, 4, 5, 9$

$6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9, 6, 6, 3, 9, 3, 3, 6, 9$

$7, 7, 5, 3, 8, 2, 1, 3, 4, 7, 2, 9, 2, 2, 4, 6, 1, 7, 8, 6, 5, 2, 7, 9$

$8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9$

$9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9$

It has been conjectured that each natural number not equal to 216 can be written in the form $p + Tx$, where $p$ is $0$ or a prime, and $Tx=x(x+1)/2$ is a triangular number.[2]

In base 10, it is a Harshad number.

There are 216 fixed hexominoes, the polyominoes made from 6 squares.

216 is a Friedman number.

216 is the only number n, for which n-3,n-2,n-1,n+1,n+2,n+3 are all semiprimes.