Using Fermat's Little Theorem, solve the congruence 2 * x^(425) + 4 * x^(108) - 3 * x^2 + x-4 -=0 mod 107.Write your answer as a set of congruence classes modulo 107, such as {1,2,3}.

tabita57i 2020-11-27 Answered

Using Fermat's Little Theorem, solve the congruence 2x425+4x1083x2+x40 mod 107.
Write your answer as a set of congruence classes modulo 107, such as {1,2,3}.

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Expert Answer

Jaylen Fountain
Answered 2020-11-28 Author has 170 answers

Step 1
Fermat's Little Theorem says that if p is a prime number and a is an integer such that p does not divides a, then ap11modp. We need to find the solutions for 2·x425+4·x1083·x2+x40 mod 107.
It can be seen clearly that 107 can not be the answer to the given congruence, as otherwise, we will have 107 lambda−4 on the left side which is clearly not 0 mod 107. Therefore, from Fermat's Little Theorem, it follows that
xp1=x1061 mod 107.
Taking p = 107, we have
x1061 mod 107
x108=x2x106
x108x2 mod 107
We also have
x1061 mod 107
x424=x4106
x4241 mod 107
x425x mod 107
Step 2
Now, we can write the given congruence as
2·x+4·x23·x2+x40 mod 107.
x2+3x40 mod 107
This given us (x1)(x+4)0 mod 107
Solving this congruence using the basic procedure, we get the solutions x=1, 103. Therefore, the required set of congruence classes is {1, 103}.

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Jeffrey Jordon
Answered 2021-11-14 Author has 2070 answers

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