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Section1.4Exercises

2

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

3

Prove that \(2^n>n\) for all integers \(n\geq 0\) by induction.

4

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

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

6

What is the conductor for \(\{3,5\}\) or \(\{4,5\}\)? 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 \(y\).

10

Get a Sage account somewhere, such as at the SageMath Cloud or a Sage notebook server on your campus, if you don't already have one.