Eleven, mathematically

Eleven is the smallest two-digit prime, and it reads the same from either end. That makes it a palindromic prime of a rare kind: the only one with an even number of digits.

It is also written with ones alone, and it is the smallest such number that is prime. The next is 1,111,111,111,111,111,111, nineteen ones long. An American mathematician, Oscar Hoppe, proved it prime in 1916, decades before anyone had a computer to check.

Its multiples are easy to spot, however long the number. Add up the digits in the odd places (first, third, fifth…) and, separately, the digits in the even places. If the difference between the two sums divides by 11, and zero counts, so does the whole number. Take 918,082:

918082

The gold digits add up to 9 + 8 + 8 = 25, the violet ones to 1 + 0 + 2 = 3, and 25 − 3 = 22, which is 2 × 11. So 918,082 divides by 11: it is 11 × 83,462.

There is more where this came from. The mathematician Tanya Khovanova keeps a page of things that are true only of 11 on her site Number Gossip.