Final Solutions

# Final Solutions - FINAL EXAM COMMENTS AND SOLUTIONS 1...

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FINAL EXAM COMMENTS AND SOLUTIONS 1. Comments 1. This was problem 5 on Homework 3. 2. This is a variation on problem 5 of Homework 3. (We have seen operations with this property before; see e.g. problem 2 on Homework 3.) 3. This is like problems 1-4 on Homework 3 and problem 2 on Midterm 1. 4. This is like problems 1-4 on Homework 3 and problem 2 on Midterm 1. 5. This is like problems 1-3 on Homework 7. [The idea: it is required that f be invertible, but that f not be its own inverse. From there you can just look for examples.] 6. This was discussed in class on November 21, and is related to problem 1 on Homework 10. 7. This was discussed in the last week of class: ax + by = c has infinitely many solutions if and only if c is a multiple of gcd( a, b ). 8. This could be done using the Euclidean algorithm (done in class several times) or by any other method you can think of. 9. This was problem 3 on Midterm 2. 10. This is related to problem 5 on Homework 6, and problem 2 on Midterm 2.

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