Chapter 2 – Additional Exercises



Chapter 6 – Additional Exercises

1. Find a solution in integers to the following equations:

a. 1479x + 272 y = gcd(1479, 272)

b. 142785821x + 1320979y = gcd(142785821, 1320979)

c. 320827x + 1101143y = gcd(320827, 1101143)

2. Describe all integer solutions to each equation in exercise # 1.

3. Show that the gcd(n, n+1) = 1 for every integer n.

4. If [pic] and [pic] and [pic], prove that [pic].

5. If [pic] and [pic], prove that [pic] and [pic].

Hint: If [pic], then [pic] for some integer t. Then [pic]. Since [pic], there exists integers x and y where [pic]. Substitute [pic]

into [pic] and form a linear combination between a and b and recall the fact about what linear combination the gcd represents. To show [pic], assume [pic]. Use the fact that [pic], [pic], and that [pic] to show [pic]. What does this now contradict concerning a and b?

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