Find the remainder when 217054 is divided by 17.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Problem Statement:**
Find the remainder when \(2^{17054}\) is divided by 17.
**Solution Approach:**
To solve this problem, you can use Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then:
\[ a^{p-1} \equiv 1 \pmod{p} \]
In this case, \(p = 17\) and \(a = 2\).
Thus, by Fermat's Little Theorem:
\[ 2^{16} \equiv 1 \pmod{17} \]
Now, find \(17054 \mod 16\) to reduce the exponent:
\[ 17054 \div 16 = 1065 \text{ remainder } 14 \]
Thus, \(17054 \equiv 14 \pmod{16}\).
Therefore:
\[ 2^{17054} \equiv 2^{14} \pmod{17} \]
Now calculate \(2^{14} \mod 17\):
\[ 2^4 = 16,\ 16 \equiv -1 \pmod{17} \]
\[ 2^8 = (2^4)^2 = 256,\ 256 \equiv 1 \pmod{17} \]
\[ 2^{14} = 2^8 \times 2^4 = 1 \times (-1) = -1 \]
Thus:
\[ 2^{14} \equiv -1 \equiv 16 \pmod{17} \]
So, the remainder when \(2^{17054}\) is divided by 17 is 16.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a4399a9-5724-42c8-89a0-9bc27dd1a0f2%2F4cfcf277-46ae-41c0-af27-05e682a6180c%2Fqcan20d_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the remainder when \(2^{17054}\) is divided by 17.
**Solution Approach:**
To solve this problem, you can use Fermat's Little Theorem, which states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then:
\[ a^{p-1} \equiv 1 \pmod{p} \]
In this case, \(p = 17\) and \(a = 2\).
Thus, by Fermat's Little Theorem:
\[ 2^{16} \equiv 1 \pmod{17} \]
Now, find \(17054 \mod 16\) to reduce the exponent:
\[ 17054 \div 16 = 1065 \text{ remainder } 14 \]
Thus, \(17054 \equiv 14 \pmod{16}\).
Therefore:
\[ 2^{17054} \equiv 2^{14} \pmod{17} \]
Now calculate \(2^{14} \mod 17\):
\[ 2^4 = 16,\ 16 \equiv -1 \pmod{17} \]
\[ 2^8 = (2^4)^2 = 256,\ 256 \equiv 1 \pmod{17} \]
\[ 2^{14} = 2^8 \times 2^4 = 1 \times (-1) = -1 \]
Thus:
\[ 2^{14} \equiv -1 \equiv 16 \pmod{17} \]
So, the remainder when \(2^{17054}\) is divided by 17 is 16.
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