NTIC Exercises

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 $a+c\mid b+d\text{.}$

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3.

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

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4.

Prove, by induction, that if $c$ divides integers $a_i$ and we have other integers $u_i\text{,}$ then $c\mid \sum_{i=1}^n a_iu_i\text{.}$

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Exploring the conductor question is a fun way to do new math where you don't already know the answer!

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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 $\{3,4\}\text{.}$

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6.

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

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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?

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8.

What is the largest number $d$ which is a divisor of both 60 and 42?

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9.

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

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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.