Elementary Number Theory Problems 4.3 Solution (David M. Burton's 7th Edition) - Q9

My Solution for "Find the remainder when $4444^{4444}$ is divided by $9$ . [Hint: Observe that $2^{3} \equiv - 1 (\mod 9)$ .]"

Ran

16 Apr 2024

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Background

All theorems, corollaries, and definitions listed in the book's order:

I will only use theorems or facts that are proved before this question. So, you will not see that I quote theorems or facts from the later chapters.

Question

Find the remainder when $4444^{4444}$ is divided by $9$ .
[Hint: Observe that $2^{3} \equiv - 1 (\mod 9)$ .]

Solution

\begin{aligned} 4444^{4444} & \equiv (4 + 4 + 4 + 4)^{4444} By Theorem 4.5 \\ \equiv 16^{4444} \\ \equiv (- 2)^{4444} \\ \equiv 2^{4444} \\ \equiv (2^{3})^{1481} \cdot 2 \\ \equiv - 1 \cdot 2 \\ \equiv 2 \\ \equiv 7 (\mod 9) \end{aligned}

< Chapter 4.3, Q8 Chapter 4.3, Q10 >

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Elementary Number Theory Problems 4.3 Solution (David M. Burton's 7th Edition) - Q9

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