1.
Prove some or all of the facts in Proposition 1.2.8.
Exercises 1.4 Exercises
2.
Find a counterexample to show that when \(a\mid b\) and \(c\mid d\text{,}\) it is not necessarily true that \(a+c\mid b+d\text{.}\)
3.
Prove using induction that \(2^n>n\) for all integers \(n\geq 0\text{.}\)
4.
Prove, by induction, that if \(c\) divides integers \(a_i\) and we have other integers \(u_i\text{,}\) then \(c\mid \sum_{i=1}^n a_iu_i\text{.}\)
Exercise Group.
Exploring the conductor question is a fun way to do new math where you don’t already know the answer!
5.
Write up a proof of the facts from the first discussion about the conductor idea (in Section 1.1) with the pairs \(\{2,3\}\text{,}\) \(\{2,4\}\text{,}\) and \(\{3,4\}\text{.}\)
6.
What is the conductor for \(\{3,5\}\) or \(\{4,5\}\text{?}\) Prove these in the same manner as in the previous problem.
7.
Try finding a pattern in the conductors. Can you prove something about it for at least certain pairs of numbers, even if not all pairs?
8.
What is the largest number \(d\) which is a divisor of both 60 and 42?
9.
Try to write the answer to the previous problem as \(d=60x+42y\) for some integers \(x\) and \(y\text{.}\)
10.
Get a Sage worksheet account somewhere, such as at
https://cocalc.com (CoCalc) or at a Sage notebook or Jupyterlab server on your campus, if you don’t already have one.11.
Color the ‘Paint by Numbers’ FoxTrot Sunday comic 5 from 2006.
Shareable version available online, in lower resolution at
licensing.andrewsmcmeel.com/features/ft?date=2006-04-09, or see the online version of the text for a non-shareable version in higher resolution.
