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)
Exercise 4.9.29: Solve the following systems of congruences, or
state that there is no solution. Be sure to state if there are
multiple solutions.
a. {6 = 13a + b(mod 26), 13 = 4a + b(mod 26)
b. {14 = 17a + b(mod 26), 8 = 7a + b(mod 26)
c. {1 = 15a + b(mod 26), 10 = 9a + b(mod 26)
Solve a system of equations:
1-
2x = 5 mod 15
3x = 1 mod 4
2-
x = 5 mod 15
x = 2 mod 12
(Hint: Note that 15 and 12 are not relatively prime. Use the
Chinese remainder
theorem to split the last equation into equations modulo 4 and
modulo 3)
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.
Determine the solution of the following equation mod N.
1.7x≡2 mod 15, where N= 15
2.x≡8 mod 11, x≡3 mod 19, where N= 209
3.x≡2 mod 7, x≡2 mod 11, x≡1 mod 13, where= 1001