64 (number) 64 (number) 64 (sixty-four) is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number that is both a perfect square and a perfect cube. Since
Square-free integer if and only if all abelian groups of order "n" are isomorphic, which is the case if and only if any such group is cyclic. This follows from the classification of finitely generated abelian groups. A integer "n" is square-free if and only if the factor ring Z / "nZ (see modular arithmetic) is a product of fields. This follows from the Chinese remainder theorem and the fact that a ring of the form Z / "kZ is a field if and only if "k" is a prime. For every positive integer "n", the set of all positive divisors of