Question

1. Which of [0], [1], [2], [3] is equal to$\left[5^{2000}\right]$ in ${\mathrm{\mathbb{Z}}}_n$?
2. Which of [0], [1], [2], [3], [4] is equal to$\left[4^{2001}\right]$ in ${\mathrm{\mathbb{Z}}}_n$?

Short answer

The term $\left[5^{2000}\right]$ in ${\mathrm{\mathbb{Z}}}_n$ is equal to [1].

b) The term $\left[4^{2001}\right]$ in ${\mathrm{\mathbb{Z}}}_n$ is equal to [4].

Step-by-step solution

Statement of Theorem 2.2 and 2.3

Theorem 2.2 states that when$a\equiv b\left(modn\right)$and role="math" localid="1646377027618" $c\equiv d\left(modn\right)$then,

$$\begin{gathered} a+c\equiv b+d\left(modn\right) \\ ac\equiv bd\left(modn\right) \end{gathered}$$

Theorem 2.3 states that $a\equiv c\left(modn\right)$such that if $\left[a\right]=\left[c\right]$

Determine which terms are equal to 52000  in  ℤn

a)

Let the expression be $\left[5^{2000}\right]$.

It is known that $5\equiv 1\left(mod4\right)$.

Use Theorem 2.2to obtain as follows:

$$\begin{gathered} 5^2\equiv 1·1\left(mod4\right) \\ {5}^n\equiv 1^n\left(mod4\right) \end{gathered}$$

Follow the same procedure to attain that role="math" localid="1646378193570" $\left[5^{2000}\right]\equiv 1\left(mod4\right)$.

Use Theorem 2.3to obtain as follows:

$\left[5^{2000}\right]=\left[1\right]$

Thus, the term $\left[5^{2000}\right]$ in ${\mathrm{\mathbb{Z}}}_n$ is equal to [1].

Determine which terms are equal to 42001  in  ℤn

b)

Let the expression be $\left[4^{2001}\right]$.

It is known that
role="math" localid="1646379451089" $$\begin{gathered} 4^{5-1}\equiv {\left(4^4\right)}^{500} \\ {1}^{500}\equiv 4 \end{gathered}$$

UseTheorem 2.2to obtain as follows:

$\begin{array}{rcl}{4}^{2001}& \equiv & {\left({\left(4\right)}^4\right)}^{500}·4\\ & \equiv & 1^{500}·4\\ & \equiv & 4\left(mod5\right)\end{array}$

UseTheorem 2.3to obtain as follows:

$\left[4^{2001}\right]=\left[4\right]$

Thus, the term $\left[4^{2001}\right]$in is equal to [4].