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Exercises 1.4 Exercises


Find a counterexample to show that when \(a\mid b\) and \(c\mid d\text{,}\) it is not necessarily true that \(a+c\mid b+d\text{.}\)


Prove using induction that \(2^n>n\) for all integers \(n\geq 0\text{.}\)


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!


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 \(\{3,4\}\text{.}\)


What is the conductor for \(\{3,5\}\) or \(\{4,5\}\text{?}\) Prove these in the same manner as in the previous problem.


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?


What is the largest number \(d\) which is a divisor of both 60 and 42?


Try to write the answer to the previous problem as \(d=60x+42y\) for some integers \(x\) and \(y\text{.}\)


Get a Sage worksheet account somewhere, such as at (CoCalc) or at a Sage notebook or Jupyterlab server on your campus, if you don’t already have one.