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
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using euler theorem 

**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 \(\boxed{12}\).
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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