Question

Using Chinese Remainder Theorem solve for X:

x = 2 (mod 3)

x = 4 (mod 5)

x = 5 (mod 8)

I have the answer the professor gave me, but I can`t understand what`s going on. So if you could please go over the answer and explain, it would help a lot.

x = 2 (mod 3)

x = 3a + 2

3a + 2 = 4 (mod 5)

(2) 3a = 2 (2) (mod 5) -----> why number 2?

a= 4 (mod 5) ---------> What happened with 6?

a = 5b + 4

x = 3 (5b + 4) + 2 --------> what`s going on here?

x = 15 b + 14

15b + 14 = 5 (mod 8)

7b + 6 = 5 (mod 8)

7b = -1 (mod 8)

(7) 7b = 7 (mod 8) (7)

b = 1 (mod 8)

b = 8c + 1

x = 15 (8c + 1) + 14

x + 120c + 29

x = 29 (mod 120)

Answer #1

solve the system of the congruences using Chinese Remainder.
x≡2 (mod 11)
x≡8 (mod 13)
x≡5 (mod 16)

(20 pts) Use construction approach in the proof of the Chinese
Remainder Theorem to solve system of congruences ? ≡ 2 (mod 3) and
? ≡ 1 (mod 4) and ? ≡ 3 (mod 5).

Solve the system of congruences:
x = 1 (mod 3)
x = 2 (mod 4)
x = 2 (mod 5)

(a) Solve
x ≡ 11 (mod 12), x ≡ 4 (mod 5), x ≡ 0 (mod 7)
(b) Find all the solutions of the following system:
x ≡ 5 (mod 6), x ≡ 4 (mod 11), x ≡ 3 (mod 17).

solve the following set of simultaneous congruence x=1
mod (2) x= 2 mod(3) x =3:mod (5) x= 4 mod (11)

Solve the system of congruences
3x+4= 2 (mod 7)
x-13= 24 (mod 29)

Solve the following systems of congruences.
x ≡ 2 mod 3
x ≡ 3 mod 4

5.
(a) Find the values of 2k mod 9 where k = 1, 2, 3, 4, 5,
6, 7, 8, 9.
(b) Find the remainder of 2271 when this number is
divided by 9.
(c) Find the remainder of 2271 when this number is
divided by 5.
PLEASE GIVE RIGHT ANSWERS AND GOOD DETAILS ON HOW TO GET THE
ANSWERS
THE ANSWERS I GOT YESTERDAY FROM YOUR GUYS WERE WRONG!!!

Solve the linear congruence
x = 2 mod (7)
x = 1 mod (3)

Use the remainder theorem to find the remainder when f(x) is
divided by x+3. Then use the factor theorem to determine whether
x+3 is a factor of (x).
The remainder is ________
Is x+3 a factor of f(x)=3x^6-27x^4+x^3-7

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