Find the remainder when 819054 is divided by 13.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find the remainder when \(8^{19054}\) is divided by 13.
**Explanation:**
This problem involves finding the remainder of a large power when divided by a number. It can often be solved using modular arithmetic techniques such as Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \ (\text{mod} \ p)\).
Here, since 13 is a prime number, we can apply this theorem. Specifically:
1. Verify that 13 is a prime and does not divide 8.
2. Apply Fermat's Little Theorem:
\[
a^{p-1} \equiv 1 \ (\text{mod} \ p)
\]
For \(a = 8\) and \(p = 13\):
\[
8^{12} \equiv 1 \ (\text{mod} \ 13)
\]
3. Break down \(8^{19054}\) by expressing its exponent in terms of multiples of 12.
Please refer to mathematical methods such as modular arithmetic or Fermat's Little Theorem for further steps in solving such problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a4399a9-5724-42c8-89a0-9bc27dd1a0f2%2F1371bbb0-42c5-4504-9936-e4237b3455d5%2Fc5m04p_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the remainder when \(8^{19054}\) is divided by 13.
**Explanation:**
This problem involves finding the remainder of a large power when divided by a number. It can often be solved using modular arithmetic techniques such as Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \ (\text{mod} \ p)\).
Here, since 13 is a prime number, we can apply this theorem. Specifically:
1. Verify that 13 is a prime and does not divide 8.
2. Apply Fermat's Little Theorem:
\[
a^{p-1} \equiv 1 \ (\text{mod} \ p)
\]
For \(a = 8\) and \(p = 13\):
\[
8^{12} \equiv 1 \ (\text{mod} \ 13)
\]
3. Break down \(8^{19054}\) by expressing its exponent in terms of multiples of 12.
Please refer to mathematical methods such as modular arithmetic or Fermat's Little Theorem for further steps in solving such problems.
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