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 ŒaŒb D Œ1 and ŒaŒc 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.

### Algebra. Abstract and concrete by Goodman F.M.

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