# 3.E Exercises

- Page ID
- 7526

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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 8^{1789 } is divided by 28.

## Exercise \(\PageIndex{6}\):

Find the remainder,

- When \(3^{1798}\) is divided by \(28.\)
- When \(2^{1798}\) is divided by \(28.\)
- 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 \(n^3-n.\)

**Answer**-
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## Exercise \(\PageIndex{9}\)

Compute the last two digits of \(9^{1600}\).

**Answer**-
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## Exercise \(\PageIndex{10}\)

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

**Answer**-
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## Exercise \(\PageIndex{11}\)

Let \(a\) be an odd integer. Show that \(a^2 \equiv 1( \mod 8)\).

**Answer**-
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