Question

Solve each of the following congruences or explain why it is not solvable:

(i) x^{2} + 5x + 7 ≡ 0(mod 41).

(ii) x^{2} ≡ 11(mod 125)

Answer #1

For each of the following quadratic congruences, decide whether
or not the equation is solvable. If the equation is solvable, state
the number of incongruent solutions. Note that you don’t need to
actually find the solutions.
(i) x2 ≡ 47(mod 1260).
(ii) 13x2 ≡ 17(mod 1176).

Which of the following congruences have solutions?
a) x2 Congruent 7(mod 53)
b) x2 Congruent 14(mod 31)
c) x2 Congruent 53(mod 7)
d) x2 Congruent 25(mod 997)
All of them please!!

For each of the following congruences if there is a solution,
express the solution in the form x
≡ some_number (mod
some_modulus), e.g. x ≡ 6 (mod 9). To standardize
answers, some_number should always be a value
in the range {0, 1, 2, ..., some_modulus -1}. For example
x ≡ 5 (mod 8) is OK but x ≡ 13 (mod 8) is not.
If there is no solution say "No solution". You don't have to
show work for any of the problems. Type...

Solve the following equations using step-by-step equational
reasoning, and list each step.
a) Solve the following equation for x ∈ ℤ/43ℤ: x2 ≡
25 (mod 43)
b) Solve the following equation for x ∈ ℤ/19ℤ: x2 ≡ 5
(mod 19)
c) Solve the following equation for x ∈ ℤ/55ℤ: x2 ≡ 1
(mod 55)
d) Solve the following equation for x ∈ ℤ/41ℤ: 3 * x2
≡ 7 (mod 41)
e) Solve the following equation for x ∈ ℤ/49ℤ: x2...

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

Number Theory: Please show all work.
Solve each of the following equations for the unknown variable X
≡ 0, 1, 2, 3, 4, 5, 6 mod 7.
(i) 2X + 5 ≡ 6 mod 7.
(ii) 3X + 5 ≡ 6 mod 7.

i) lim x→3 (4x − 12)/x2-5x+6
ii)lim x→+∞ (e2x-1-x)/x2
iii)limx→0 (sinh(x2))/ cosh x-1
iv) limx→0 sin2x/sin7x

For each part, find all possible solutions to the given
equations (if any exist).
i ) x = 7 mod 9, x =3 mod 4
ii) 3x^2 +1 = 0 mod 10
iii) 3x+7 = 9 mod 10

Let
t= 20389208 mod 4 and M= t+25. Find the following,
a. (i) 2^M mod 7; (ii) 10^M mod 7; (iii) 3^M mod 7
b. 20389208 mod m for m= 2,3,5,9 and 11
c. 123456789987654321 mod m for m= 2,3,4,5,9 and 11

1) What is a function? Give two examples
2) Solve x2−4=0 factorize
3) Graph following line y= -3x+1
4) Find the domain of following functions
y= x3+x2−5x+10
Y=5X−1
5) Find the following limits
limn→∞ n32n3
limn→2 1000n
Please provide all the steps!

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