Square-free integer
In mathematics, a square-free, or quadratfrei (from German language) integer, is an integer which is divisible by no other perfect square than 1. For example, 10 is square-free but 18 is not, as 18 is divisible by 9 = 32. The smallest positive square-free numbers are
The radical of an integer is its largest square-free factor. An integer is square-free if and only if it is equal to its radical.
Any arbitrary positive integer n can be represented in a unique way as the product of a powerful number and a square-free integer, which are coprime. The square-free factor is the largest square-free divisor k of n that is coprime with n/k.
Any arbitrary positive integer n can be represented in a unique way as the product of a square and a square-free integer :
In this factorization, m is the largest divisor of n such that m2 is a divisor of n.
Every prime number is square-free.
A positive integer n is square-free if and only if in the prime factorization of n, no prime factor occurs with an exponent larger than one. Another way of stating the same is that for every prime factor p of n, the prime p does not evenly divide n / p. Also n is square-free if and only if in every factorization n = ab, the factors a and b are coprime. An immediate result of this definition is that all prime numbers are square-free.
A 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.
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