Question

Show that every positive integer can be written as the product of a square (possibly 1) and a square-free integer. (a square-free integer is an integer that is not divisible by any perfect squares other than one). Can some one prove this and also do an example? Like the integer 36

Answer 1

The p's are the prime factorization, right? Then you are done. Multiply the perfect squares to get one factor and multiply the square free parts to get the other factor.

Answer 2

what about the number 36? the prime factorization would be 2^2 * 3^2 and there is no square-free part?

Answer 3

36/1. doesn't 1 work as a square-free integer since it's only divisible by 1?

Answer 4

so that's to say 1 is both a square integer and a square-free integer? that's confusing...