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# 1E: Exercises

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##### Exercise $$\PageIndex{1}$$
1. For every positive integer $$n$$, prove that $$\frac{1}{(1)(2)}+\frac{1}{(2)(3)}+\cdots +\frac{1}{(n)(n+1)}=\frac{n}{n+1}$$.

2.
Answer

We can rewrite each term as $$\frac{1}{k(k+1)} = \frac{k+1 - k}{k(k+1)} = \frac{1}{k} - \frac{1}{k+1}$$. Therefore, the sum telescopes as follows: \begin{align*} &\frac{1}{1} - \frac{1}{2} + \frac{1}{2} - \frac{1}{3} + \cdots + \frac{1}{n} - \frac{1}{n+1} \\ &= 1 - \frac{1}{n+1} = \frac{n}{n+1}. \end{align*}

##### Exercise $$\PageIndex{2}$$
1. For any positive integer n, prove that $$2^n3^{2n}-1$$ is always divisible by $$17$$.

2.
Answer

By Fermat's Little Theorem, $$2^{16} \equiv 1 \pmod{17} and 3^{16} \equiv 1 \pmod{17}$$. Therefore, $$2^{16n} \equiv 1 \pmod{17}$$ and $$3^{32n} \equiv 1 \pmod{17}$$. Hence, $$2^n3^{2n} - 1 \equiv 1 \cdot 1 - 1 \equiv 0 \pmod{17}$$.

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

a. Let $$a, b, c \in \mathbb{Z}_+$$.  Show that $$\gcd(a, bc) = 1$$ if and only if $$\gcd(a, b) = 1$$ and $$\gcd(a, c) = 1$$.

b. Let $$a,b,c,d \in \mathbb{Z}.$$  If $$a|b$$ and $$c|d$$, show that $$gcd(a,c)|gcd(b,d).$$

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

Show that $$a^5 \equiv a \pmod{5}$$ for all integers $$a$$.

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

Using Euclidean algorithm to find $$\gcd(2520,154)$$ and express $$\gcd(2520, 154)$$ as an integer combination of $$2520$$ and $$154$$. Also,

Using the Euclidean algorithm to find $$\gcd(-2520,154)$$ and express $$\gcd(-2520, 154)$$ as an integer combination of $$-2520$$ and $$154$$.

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

For every positive integer $$n$$, prove that $$n^3-n$$ is divisible by $$3$$.

##### Exercise $$\PageIndex{7}$$

For any $$k \in \mathbb{N}$$ prove that $$\gcd(4k+3, 7k+5)=1$$.

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

a) Use the Euclidean algorithm to find the $$\gcd(-29,571)$$?

b) Find integers $$x$$ and $$y$$ s.t. $$\gcd(-29,571)= -29(x)+571(y)$$?

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

Show that any two consecutive odd integers are relatively prime.

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

If $$m,n$$ are any odd integers, show that $$m^2-n^2$$ is divisible by $$8.$$

##### Exercise $$\PageIndex{11}$$

a) Prove that for all integers $$n\geq 1,$$

$$\frac{1}{2!}+ \frac{2}{3!} + \cdots +\frac{n}{(n+1)!} = 1-\frac{1}{(n+1)!}.$$

b) Prove that for all integers $$n\geq 1,$$

$$\frac{1}{\sqrt{1}}+ \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{n}.$$

##### Exercise $$\PageIndex{12}$$

For integer $$n\geq 1$$, define

$$S_n=- {2 \choose 2}+{3 \choose 2}-{4 \choose 2}+{5 \choose 2}\cdots + {2n+1 \choose 2}.$$

1. Evaluate $$S_n$$ for $$n=1,2,3,4$$ and $$5.$$

2. Use part (a) to guess a formula for $$S_n.$$

3. Use mathematical induction to prove your guess.

##### Exercise $$\PageIndex{13}$$

Find the remainder when $$8^{391}$$ is divided by $$5.$$

##### Exercise $$\PageIndex{14}$$

For each of the following pairs of integers $$a$$ and $$n.$$ show that $$a$$ and $$n$$ are relatively prime, determine multuplicative inverse of $$[a]$$ in $$\mathbb{Z}_n,$$ and Find all integers $$x$$ for $$ax \cong 11 (mod \, n).$$

1.  $$a=16, n=35.$$
2. $$a=69, n=89.$$

This page titled 1E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.

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