Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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using euler theorem
![**Problem:**
Find the remainder when \(8^{19054}\) is divided by 13.
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When tackling a problem like this, you can use Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is any integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \ (\text{mod} \ p)\).
Here, \(a = 8\) and \(p = 13\), thus \(8^{12} \equiv 1 \ (\text{mod} \ 13)\).
To find \(8^{19054} \ (\text{mod} \ 13)\), we first determine the exponent modulo 12:
\[19054 \div 12 = 1587 \text{ remainder } 10.\]
So, \(19054 \equiv 10 \ (\text{mod} \ 12)\).
Therefore, \(8^{19054} \equiv 8^{10} \ (\text{mod} \ 13)\).
Now, compute \(8^{10} \ (\text{mod} \ 13)\):
1. \(8^2 = 64 \equiv 12 \equiv -1 \ (\text{mod} \ 13)\)
2. \(8^4 = (8^2)^2 \equiv (-1)^2 \equiv 1 \ (\text{mod} \ 13)\)
3. \(8^{10} = (8^4)^2 \cdot 8^2 \equiv 1^2 \cdot (-1) \equiv -1 \equiv 12 \ (\text{mod} \ 13)\)
Thus, the remainder when \(8^{19054}\) is divided by 13 is \(\boxed{12}\).](https://content.bartleby.com/qna-images/question/2a4399a9-5724-42c8-89a0-9bc27dd1a0f2/2cd5e16b-5d55-468d-80e1-46113fc95607/8eb6zwf_processed.png)
Transcribed Image Text:**Problem:**
Find the remainder when \(8^{19054}\) is divided by 13.
---
When tackling a problem like this, you can use Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is any integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \ (\text{mod} \ p)\).
Here, \(a = 8\) and \(p = 13\), thus \(8^{12} \equiv 1 \ (\text{mod} \ 13)\).
To find \(8^{19054} \ (\text{mod} \ 13)\), we first determine the exponent modulo 12:
\[19054 \div 12 = 1587 \text{ remainder } 10.\]
So, \(19054 \equiv 10 \ (\text{mod} \ 12)\).
Therefore, \(8^{19054} \equiv 8^{10} \ (\text{mod} \ 13)\).
Now, compute \(8^{10} \ (\text{mod} \ 13)\):
1. \(8^2 = 64 \equiv 12 \equiv -1 \ (\text{mod} \ 13)\)
2. \(8^4 = (8^2)^2 \equiv (-1)^2 \equiv 1 \ (\text{mod} \ 13)\)
3. \(8^{10} = (8^4)^2 \cdot 8^2 \equiv 1^2 \cdot (-1) \equiv -1 \equiv 12 \ (\text{mod} \ 13)\)
Thus, the remainder when \(8^{19054}\) is divided by 13 is \(\boxed{12}\).
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