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In mathematics, a square-free integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32. The small square-free numbers are 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 1717 seventeen is the natural number following 16 and preceding 18. In mathematics Seventeen is the 7th smallest prime number, and is a Fermat prime. The next prime is nineteen, with which it comprises a twin prime. 17 is the sum of the first four primes., 1919 nineteen is the natural number following 18 and preceding 20. In mathematics Nineteen is the 8th smallest prime number. The sequence continues 23, 29, 31, 37. 19 is the seventh Mersenne prime exponent. It is also a Keith number, because it recurs in a, 212 1 2 1 21 twenty- one is the natural number following 20 and preceding 22. Cardinal twenty-one Ordinaltwenty-first Factorization Roman numeralXXI Binary10101 Hexadecimal15 In mathematics Twenty-one is a Fibonacci number, a Harshad number, a Motzkin numbe, 222 2 2 2 22 twenty- two is the natural number following 21 and preceding 23. Cardinal twenty-two Ordinaltwenty-second Factorization Roman numeralXXII Binary10110 Hexadecimal16 In mathematics Twenty-two is a composite number, its proper divisors being 1, 2, 232 3 23 twenty-three is the natural number following 22 and preceding 24. Cardinal twenty-three Ordinaltwenty-third Factorization prime Roman numeralXXIII Binary10111 Hexadecimal17 In mathematics Twenty-three is the ninth prime number, the sequence continu, 262 6 2 6 26 twenty-six is the natural number following 25 and preceding 27. Cardinal twenty-six Ordinaltwenty-sixth Factorization Roman numeralXXVI Binary11010 Hexadecimal1A In mathematics Twenty-six is a composite number, its proper divisors being 1, 2, a, 292 9 29 twenty-nine is the natural number following 28 and preceding 30. Cardinal twenty-nine Ordinaltwenty-ninth Factorization prime Roman numeralXXIX Binary11101 Hexadecimal1D In mathematics It is the tenth prime number, and also a primorial prime, the n, 3030 thirty is the natural number following 29 and preceding 31. Cardinalthirty Ordinalthirtieth Numeral system Factorization Divisors2, 3, 5, 6, 10, 15 Roman numeralXXX Binary11110 Hexadecimal1E In mathematics It is a primorial as well as the sum of the sq, 31, 33, ... 1 Equivalent characterizations of square-free numbers
The integer n is square-free iff in the prime factorization of n, no prime number occurs more than once. Another way of stating the same is that for every prime divisor p of n, the prime p does not divide n / p. Yet another formulation: n is square-free iff in every factorization n=ab, the factors a and b are coprime.
The positive integer n is square-free iff μ(n) ≠ 0, where μ denotes the Möbius function.
The positive integer n is square-free if and only if all abelian groups of order n are isomorphic, which is the case if and only if all of them are cyclic. This follows from the classification of finitely generated abelian groups.
The integer n is square-free iff 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 n becomes a partially ordered set if we use divisibility as the order relation. This partially ordered set is always a distributive lattice. It is a Boolean algebra if and only if n is square-free.
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