Using the Chinese remainder theorem solve for x:
x = 1 mod 3
x = 5 mod 7
x = 5 mod 20
Please show the details, I`m trying to understand how to solve
this problem since similar questions will be on my exam.
a. Solve 7x + 5 ≡ 3 (mod 19).
b. State and prove the Chinese Remainder Theorem
c. State and prove Euler’s Theorem.
d. What are the last three digits of 9^1203?
e. Identify all of the primitive roots of 19.
f. Explain what a Feistel system is and explain how to decrypt something encoded with a Feistel system. Prove your result.
Use the remainder theorem to find the remainder when f(x) is
divided by x-1. Then use the factor theorem to determine whether
x-1 is a factor of f(x).
f(x)=4x4-9x3+14x-9
The remainder is ____
Is x-1 a factor of f(x)=4x4-9x3+14x-9?
Yes or No
The Chinese Remainder Theorem for Rings.
Let R be a ring and I and J be ideals in R such that I + J = R.
(a) Show that for any r and s in R, the system of equations x ≡ r
(mod I) x ≡ s (mod J) has a solution. (b) In addition, prove that
any two solutions of the system are congruent modulo I ∩J. (c) Let
I and J be ideals in a ring R...
3. Find the quotient and remainder using long division. x3 + 7x2
− x + 1 x + 8
quotient = ?
remainder = ?
4. Simplify using long division. (Express your answer as a
quotient + remainder/divisor.)
f(x) =
8x2 − 6x + 3
g(x) = 2x
+ 1
5.
Find the quotient and remainder using long division.
9x3 + 3x2 + 22x
3x2 + 5
quotient
remainder
6.
Use the Remainder Theorem to evaluate
P(c).
P(x) = x4...