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The prime factorization of a positive integer is its expression as a product of primes Using the “tree method” factor the following numbers into their prime factors and check them using the calculator program or the internet tool located here: Prime factorizations can be used to compute the number of divisors of a positive integer, as well as the sum of its divisors.

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Th e prime factorization of a composite number is the number written as a product of its prime factors Two numbers are relatively prime iff they have no common prime factors. You can use factor pairs and a factor tree to help fi nd the prime factorization of a number.

By problem 4a, if n(α) is prime, and we write α = βγ, either n(β) or n(γ) is 1, so either β or γ is a unit, and α is irreducible

Since n(p) = p2 has only one factorization using numbers greater than 1 (p2 = p · p), p can only be factored as the product of two irreducibles with norm p. Prime factorization is a factor string expressing a number as the product of only prime factors Every number has exactly one prime factorization This prime factorization can be written using exponents if any of its prime factors appear more than once in the string.

Lemma 2 says that every integer larger than 1 is equal to a product of primes, and hence has a prime factorization (just write down the factors in increasing order). 4 and 9 are relatively prime