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Thu 2 Mar, 2006 06:52 pm
Use the Well Ordering Principle to prove that every positive integer can be written as the product of a square and a square-free integer.
Note:
A square-free integer has no divisor bigger than 1 that is a square.
For example, 8 and 18 are NOT square-free integers because
4 divides 8 and 4 = 2^2
9 divides 18 and 9 = 3^2
I don't know what the WOP is, but if you take all of the prime factors, and put all of the pairs in one set and the leftovers in another set, the product of the first set will be a square, and the product of the second set will be square free.
Yeah, I know. The proof you're talking about uses the Fundamental Theorem of Arithmetic (Prime Power Factorization) and then some re-arranging to get the desired result.
The WOP states that every subset of the natural numbers has a least element and is thus non-empty.
I am told that the WOP can be used to prove the claim in question. I just don't see how to relate it. Thanks!
Induction is allowed by the WOP, or in reality the WOP allows induction.
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