Exercises 1.4 Exercises
¶1.
Prove some or all of the facts in Proposition 1.2.8.
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{.}
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.