111 (number)

"Number 111" redirects here. For the 1919 film, see Number 111 (1919 film). For the 1938 film, see Number 111 (1938 film). For the emergency telephone number in New Zealand, see 111 (New Zealand telephone number).

For other uses, see 111 (disambiguation).

← 110 111 112 →
← 110 111 112 113 114 115 116 117 118 119 → List of numbers Integers ← 0 100 200 300 400 500 600 700 800 900 →
Cardinal	one hundred eleven
Ordinal	111th (one hundred eleventh)
Factorization	3 × 37
Divisors	1, 3, 37, 111
Greek numeral	ΡΙΑ´
Roman numeral	CXI, cxi
Binary	11011112
Ternary	110103
Senary	3036
Octal	1578
Duodecimal	9312
Hexadecimal	6F16

111 (one hundred [and] eleven) is the natural number following 110 and preceding 112.

In mathematics

111 is the fourth non-trivial nonagonal number, and the seventh perfect totient number.

111 is furthermore the ninth number such that its Euler totient $\varphi (n)$ of 72 is equal to the totient value of its sum-of-divisors:

$\varphi (111)=\varphi (\sigma (111)).$

Two other of its multiples (333 and 555) also have the same property (with totients of 216 and 288, respectively).

Magic squares

111 is adjacent to 110 and 112, the minimal side lengths of perfect squared squares that are tiled by smaller squares of distinct side lengths.

The smallest magic square using only 1 and prime numbers has a magic constant of 111:

31	73	7
13	37	61
67	1	43

Also, a six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:

1	11	31	29	19	20
2	22	24	25	8	30
3	33	26	23	17	9
34	27	10	12	21	7
35	14	15	16	18	13
36	4	5	6	28	32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

On the other hand, 111 lies between 110 and 112, which are the two smallest edge-lengths of squares that are tiled in the interior by smaller squares of distinct edge-lengths (see, squaring the square).

Properties in certain radices

111 is $R_{3}$ or the second repunit in decimal, a number like 11, 111, or 1111 that consists of repeated units, or ones. 111 equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are repdigits of the form $3n\times 37$. As a repunit, it also follows that 111 is a palindromic number. All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in decimal, but is prime in base two, where 111₂ = 7₁₀. It is also prime in many other bases up to 128 (3, 5, 6, ..., 119) (sequence A002384 in the OEIS). In base 10, it is furthermore a strobogrammatic number, as well as a Harshad number.

In base 18, the number 111 is 7³ (= 343₁₀) which is the only base where 111 is a perfect power.

See also

• 111 is the atomic number of roentgenium.
• The number 111 is called a Nelson in cricket, famously considered unlucky by umpire David Shepherd.

References

Further reading

Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134