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Let n be a positive integer. Show that every abelian group of order n is cyclic if and only if n is not divisible by the square of any prime.

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A number that only has itself and one other factor is called a prime number. Or, to put it another way, only it and one can be divided equally.

Divided by a non-prime, does it make sense?

Solution: A prime can be produced by dividing a nonprime by another nonprime. For instance, the result of the division of 12 (not a prime number) by 4 (also not a prime number) is 3. (which is prime).

How many prime numbers precede n?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, and 97 are the first 25 prime numbers. (sequence A000040 in the OEIS). The maximum even number is two.

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