111 (number)

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 of 72 is equal to the totient value of its sum-of-divisors:

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

Magic squares

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

3173713376167143

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

111312919202222425830333262317934271012217351415161813364562832

(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 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 . 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) . 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.

111 (number)

Bron en licentie