Math 58 - Number Theory

September 7, 2006

Homework 2 (due September 12):

(Most of these exercises are from the text, App. A, 1.3, or 1.5.)

Prove rigorously using only the axioms in Appendix A:

1. If a and b are integers, then (–a)(–b) = ab.

2. If a then a² ≥ 0.

3. If a < b and c < 0 then ac > bc.

Prove by induction (or any variant on induction):

4. If n ≥ 4 then 2ⁿ < n!.

Prove by any method:

(Extreme rigor not required.)

5. If x and y are integers (different from each other) then x – y | x² – y².

(You can skip this one if you are satisfied with your proof of #7.)

6. If x and y are integers (different from each other) and n is any positive

integer, then x – y | xⁿ – yⁿ.

7. If a, b, c, d , a and c are nonzero, and a | b and c | d, then ac | bd.

Prove OR disprove:

(If the statement is false, a single example is enough to disprove it. If any of these

is true, induction might be helpful in proving it.)

8. If a is any integer, then a² – a is even.

9. If a is any integer, then a³ – a is a multiple of 3.

10. If a is any integer, then a⁴ – a is a multiple of 4.

11. If a is any integer, then a⁵ – a is a multiple of 5.

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