NTIC Exercises

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Exercises 1.4 Exercises

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1.

Prove some or all of the facts in Proposition 1.2.8.

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2.

Find a counterexample to show that when $a ∣ b$ and $c ∣ d,$ it is not necessarily true that $a + c ∣ b + d .$

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3.

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

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4.

Prove, by induction, that if $c$ divides integers $a_{i}$ and we have other integers $u_{i},$ then $c ∣ \sum_{i = 1}^{n} a_{i} u_{i} .$

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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 ${2, 3},$ ${2, 4},$ and ${3, 4} .$

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6.

What is the conductor for ${3, 5}$ or ${4, 5} ?$ 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 $d = 60 x + 42 y$ for some integers $x$ and $y .$

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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.

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11.

Color the ‘Paint by Numbers’ Sunday comic⁵ from 2006.

Shareable version available online, in lower resolution, or see the online version of the text for a non-shareable version in higher resolution.