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Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 by Frederick M. Goodman

By Frederick M. Goodman

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Additional resources for Algebra: Abstract and Concrete (Stressing Symmetry) (2.5 Edition)

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Natural numbers that are not prime are called composite; they can be written as a product of prime numbers, for example, 42 D 2 3 7. 6. 3. A natural number is prime if it is greater than 1 and not divisible by any natural number other than 1 and itself. To show formally that every natural number is a product of prime numbers, we have to use mathematical induction. 4. Any natural number other than 1 can be written as a product of prime numbers. Proof. We have to show that for all natural numbers n 2, n can be written as a product of prime numbers.

12. 18. The prime factorization of a natural number is unique. Proof. We have to show that for all natural numbers n, if n has factorizations n D q1 q2 : : : qr ; n D p1 p2 : : : ps ; where the qi ’s and pj ’s are prime and q1 Ä q2 Ä Ä qr and p1 Ä p2 Ä Ä ps , then r D s and qi D pi for all i . We do this by induction on n. First check the case n D 1; 1 cannot be written as the product of any nonempty collection of prime numbers. So consider a natural number n 2 and assume inductively that the assertion of unique factorization holds for all natural numbers less than n.

Show that X1 D fx0 ; x1 ; : : : ; xk g and X2 D X n X1 are both invariant under . 6. 6. DIVISIBILITY IN THE INTEGERS 25 the set of permutations of a collection of identical objects has an algebraic structure. We can do computations in these algebraic systems in order to answer natural (or unnatural) questions, for example, to find out the order of a perfect shuffle of a deck of cards. In this section, we return to more familiar mathematical territory. We study the set of integers, probably most familiar algebraic system.

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