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
Prove the following more general version of the Chinese
Remainder Theorem: Theorem. Let m1, . . . , mN ∈ N, and let M =
lcm(m1, . . . , mN ) be their least common multiple. Let a1, . . .
, aN ∈ Z, and consider the system of simultaneous congruence
equations x ≡ a1 mod m1 . . . x ≡ aN mod mN This system
is solvable for x ∈ Z if and...
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...
a) Use Fermat’s little theorem to compute 52003 mod 7,52003 mod 11, and 52003 mod 13.
b) Use your results from part (a) and the Chinese remaindertheorem to find 52003 mod 1001. (Note that1001 = 7 ⋅ 11 ⋅ 13.)
1. Use backward substitution to solve:
x=8 (mod 11)
x=3 (mod 19)
2. Fine the subgroup of Z24 (the operation is addition) generates by the element 20.
3. Find the order of the element 5 in (z/7z)