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# 3.E Exercises

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## Exercise $$\PageIndex{1}$$:

Let $$a, b, c \in \mathbb{Z}$$, such that $$a \equiv b (mod\,n).$$

Show that $$ac=bc(mod\,n).$$

## Exercise $$\PageIndex{2}$$:

Find the remainder when $$(201)(203)(205)(207)$$ is divided by $$13.$$

## Exercise $$\PageIndex{3}$$:

Show that the sum of 2 odd integers is even.

## Exercise $$\PageIndex{4}$$:

Given that February 14, 2018, is a Wednesday, what day of the week will February 14, 2090 be?

## Exercise $$\PageIndex{5}$$:

Find the remainder when 81789 is divided by 28.

## Exercise $$\PageIndex{6}$$:

Find the remainder,

1. When $$3^{1798}$$ is divided by $$28.$$
2. When $$2^{1798}$$ is divided by $$28.$$
3. When $$7^{5453}$$ is divided by $$8.$$

## Example $$\PageIndex{7}$$:

Given a positive integer $$x,$$ rearrange its digits to form another integer $$y.$$ Explain why $$x-y$$ is divisible by $$9.$$

## Exercise $$\PageIndex{8}$$

Prove that for all integer $$n\geq 1,\,6$$ divides $$n^3-n.$$

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## Exercise $$\PageIndex{9}$$

Compute the last two digits of $$9^{1600}$$.

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## Exercise $$\PageIndex{10}$$

Show that $$a^2+b^2 \notequiv 3( \mod 4)$$ for any integers $$a$$ and $$b$$.

## Exercise $$\PageIndex{11}$$
Let $$a$$ be an odd integer. Show that $$a^2 \equiv 1( \mod 8)$$.