MY-ASSIGNMENTEXPERT™可以为您提供ccbcmd.edu Math108 Discrete Mathematics离散数学课程的代写代考和辅导服务！
Math108课程简介
Math 108: Discrete Mathematics has been evaluated and recommended for 3 semester hours by ACE and may be transferred to over 2,000 colleges and universities. With this self-paced course, you get engaging lessons, expert instructors who make even the most challenging math and computer science topics simple, and an excellent resource for getting a head start on your degree.
Prerequisites
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Overview
Math 108: Discrete Mathematics has been evaluated and recommended for 3 semester hours and may be transferred to over 2,000 colleges and universities. With this self-paced course, you get engaging lessons, expert instructors who make even the most challenging math and computer science topics simple, and an excellent resource for getting a head start on your degree.
Math108 Discrete Mathematics HELP（EXAM HELP， ONLINE TUTOR）
Let $z \in \mathbb{Z}$. Show that the following statements are true
(1) $z^5 \bmod 5=z \bmod 5$.
To show that the statements are true, we need to use Fermat’s Little Theorem, which states that if $\$ \mathrm{p} \$$ is a prime number and $\$ \mathrm{a} \$$ is an integer that is not divisible by $\$ p \$$, then $\$ a^{\wedge}{p-1}$ \equiv $1 \backslash p m o d p \$$.
(1) For any integer $\$ z \$$, we have $z^5 \equiv\left(z^4\right) \cdot z \equiv\left(z^4 \bmod 5\right) \cdot(z \bmod 5)(\bmod 5)$
By Fermat’s Little Theorem, we know that $\$ z^{\wedge} 4$ \equiv $1 \backslash$ pmod $5 \$$ if $\$ 5 \$$ does not divide $\$ \mathrm{z} \$$. Thus, we have $z^5 \equiv z(\bmod 5)$
which proves the statement.
(2) $z^7 \bmod 7=z \bmod 7$.
(2) Similarly, for any integer $\$ z \$$, we have
$$
z^7 \equiv\left(z^6\right) \cdot z \equiv\left(z^6 \bmod 7\right) \cdot(z \bmod 7)(\bmod 7)
$$
By Fermat’s Little Theorem, we know that $\$ z^{\wedge} 6$ \equiv $1 \backslash$ \pmod $7 \$$ if $\$ 7 \$$ does not divide $\$ \mathrm{z} \$$. Thus, we have
$$
z^7 \equiv z(\bmod 7)
$$
which proves the statement.
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