Question

1. Does the Diophantine equation 12x + 33y = 0 have an integer solution? If so, can you list all integer solutions?

2. Does the Diophantine equation 12x + 33y = 1 have an integer solution? If so, can you list all integer solutions?

3. Does the Diophantine equation 12x + 33y = 9 have an integer solution? If so, can you list all integer solutions?

Answer #1

In this problem, consider the Diophantine equation: 703 = 30x +
11y.
(a) Describe ALL integer solutions.
(b) Describe ALL positive integer solutions i.e. x, y integers
with x > 0 and y > 0 making the equation true.

Use the Euclidean algorithm to find all integer solutions to the
following diophantine equation:
2x + 6y - 9z = 13
Find all positive integer solutions to the following diophantine
equation:
2x + 6y + 5z = 24

1a) Find all the integer solutions of each of the following
linear Diophantine equations:
(i) 2x + y = 2,
(ii) 3x - 4y = 0,
and
(iii) 15x + 18 y =17.
1b) Find all solutions in positive integers of each of the
following linear Diophantine equations:
(i) 2x + y = 2,
(ii) 3x - 4y = 0,
and
(iii) 7x + 15 y = 51.

4.
What is a linear Diophantine equation of two variables? How many
solutions can such an equation have? How can the solution(s) be
found?

For which integers c, 0 ≤ c < 30, does the congruence 12x ≡ c
(mod 30) have at least one solution? Give the solutions for each
such c.

how
would you solve the below diophantine equation using thw method of
congruence. - number theory
Show that the equation 11x^2 + 10x - y^2 + 2 = 0 has no
solutions.

NUMBER THEORY
1. Solve the following diophantine equation 18x-14y=8
for a general solution.
2. Solve the following congruence x+12≡ 5(mod 8) for a
general solution.

Consider the first order separable equation
y′=12x^3y(1+2x^4)^1/2. An implicit general solution can be written
in the form y=Cf(x) for some function f(x) with C an arbitrary
constant. Here f(x)= Next find the explicit solution of the initial
value problem y(0)=1
y=

Does the equation x2 + 5x≡ 13 (mod 34) have
solutions? Can you ﬁnd the solution?

Given a matrix A, the equation Ax=b might have 0 solutions for
all b, or 1 solution for all b, or 0 solutions for some choices of
b and 1 solution for others, or 0 solutions for
some choices of b and ∞ solutions for others, or 1 solution for
some b and ∞ solutions for others, or 0,1, or ∞ solutions for
different choices of b. (7 different combinations in
all.) Which of these combinations are actually possible? Justify...

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