11 (number) is divisible by eleven. This also works by adding a trailing zero instead of a leading one: 65 + 63 + 70 = 198, which is divisible by 11. This also works with larger groups of digits, providing that each group has an even number of digits (not all groups have to have the same number of digits). An easy way of multiplying numbers by 11 in base 10 is: If the number has: In base 13 and higher bases (such as hexadecimal), 11 is represented as B, where ten is A. In duodecimal, however, 11 is sometimes represented as
Prime number theorem on primes in arithmetic progressions as a special case. In the theory of finite groups the Sylow theorems imply that, if a power of a prime number formula_195 divides the order of a group, then it has a subgroup of order formula_195. By Lagrange's theorem, any group of prime order is a cyclic group, and by the Burnside theorem any group whose order is divisible by only two primes is solvable. For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications