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

tabita57i

Answered question

2020-11-27

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}.

Answer & Explanation

Jaylen Fountain

Jaylen Fountain

Skilled2020-11-28Added 169 answers

Step 1
According to Fermat's Little Theorem, if an is an integer such that p does not divide it and p is a prime number, 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}.

Jeffrey Jordon

Jeffrey Jordon

Expert2021-11-14Added 2605 answers

Answer is given below (on video)

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