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By Goodman F.M.

ISBN-10: 0979914205

ISBN-13: 9780979914201

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Define sequences jnj > n1 > n2 0 and q1 ; q2 ; : : : by induction, as follows. 6. DIVISIBILITY IN THE INTEGERS This process must stop after no more than n steps with some remainder nrC1 D 0. 10. m; n/: Proof. Write m D n 1 and n D n0 . nk 2 ; nk Ä 0 1/ 1 1 qk Ä Ä 0 1 q 1 1 The matrix Qk D is invertible with inverse Qk D k . nr a; nr b/, where a b is the first row of Q 1 . m; n/ and nr is a common divisor of m and n. 9, nr is the greatest common divisor of m and n. m; n/. 32 1. 11. Find the greatest common divisor of 1734282 and 452376.

3. Prove that multiplication in Zn is commutative and associative, and that Œ1 is an identity element for multiplication. 4. Compute the congruence class modulo 12 of 4237 . 10 constitute an experimental investigation of zero divisors and invertible elements in Zn . 5. Can an element of Zn be both invertible and a zero divisor? 6. If an element of Zn is invertible, is its multiplicative inverse unique? That is, if Œa is invertible, can there be two distinct elements Œb and Œc such that ŒaŒb D Œ1 and ŒaŒc D Œ1?

2. 9. m:n/. m; n/, then b divides y. 3. Suppose that a natural number p > 1 has the property that for all nonzero integers a and b, if p divides the product ab, then p divides a or p divides b. Show that p is prime. 16. 4. m; n/ explicitly as an integer linear combination of m and n. 5. jmj; jnj/. 6. m; n/ is the largest natural number dividing m and n. 7. m; n/ \ N. 8. m; n/. 9. Show that if p is a prime number and a is any nonzero integer, then either p divides a or p and a are relatively prime.

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Algebra. Abstract and concrete by Goodman F.M.


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