Assignment 2

Submitted on 4 September 2024.

Question 1

For integers a and b with b0, state the definition of b divides a.

Question 2

Prove that for any integers a, b, r, and s, if ab and ac, then abr+cs.

Question 3

Prove that if ab and bc, then ac.

Question 4

Suppose we have arbitrary integers , s, and d such that s. Prove that s+d if and only if d.

Question 5

Suppose a=bq+r. Prove the following equality of sets:
\{common divisors of a and b\} = \{common divisors of b and r\}.

Question 6

Disprove: If abc then ab or ac.

Question 7

Disprove: If ac and bc then abc.

Question 8

Prove: If ac and bc and (a,b)=1, then abc.
Hint: c=bt (Why?), so abt. Use Theorem 1.4.

Question 9

Use the Euclidean Algorithm to find the greatest common divisor d of 657 and 306, and express d as a Z-linear combination of 657 and 306.

Question 10

Prove that for any natural number n, 67n1.