"Find the smallest positive integer solution to the..." solved and explained - Plainmath

Albarellak

Answered question

2021-01-31

Find the smallest positive integer solution to the following system of congruences:

x \equiv 17 (mod 35)

x \equiv 8 (mod 43)

Answer & Explanation

lamusesamuset

Skilled2021-02-01Added 93 answers

Given,

x \equiv 17 (mod 35)

and

x \equiv 8 (mod 43)

Step 2
Now x = 17 + 35k, for some

k \in Z

Then,

17 + 35 k \equiv 8 (mod 43)

35 k \equiv - 9 (mod 43)

35 k \equiv 34 (mod 43)

\Rightarrow gcd (35, 43) = 1

Then, there exists x,

y \in Z

such that 35x + 43y = 1

∵ 35 = 4 \times 8 + 3

43 = 35 \times 1 + 8

8 = 3 \times 2 + 2

Step 3

3 = 2 \times 1 + 1

\Rightarrow 1 = 3 - 2 \times 1

\Rightarrow 1 = 3 - (8 - 3 \times 2) \times 1

\Rightarrow 1 = 3 \times 3 - 8 \times 1

\Rightarrow 1 = 3 \times (35 - 4 \times 8) - 8 \times 1

\Rightarrow 1 = 3 \times 35 - 8 \times 13

\Rightarrow 1 = 3 \times 35 - (43 - 35 \times 1) \times 13

\Rightarrow 1 = 16 \times 35 - 43 \times 13

\Rightarrow 1 = 35 (16) + 43 (- 13)

Hence, x = 16

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

The distance between the centers of two circles C1 and C2 is equal to 10 cm. The circles have equal radii of 10 cm.
A part of circumference of a circle is called
A. Radius
B. Segment
C. Arc
D. Sector
The perimeter of a basketball court is 108 meters and the length is 6 meters longer than twice the width. What are the length and width?
What are the coordinates of the center and the length of the radius of the circle represented by the equation $x^{2} + y^{2} - 4 x + 8 y + 11 = 0$ ?
Which of the following pairs of angles are supplementary?
128,62
113,47
154,36
108,72
What is the surface area to volume ratio of a sphere?
An angle which measures 89 degrees is a/an _____.
right angle
acute angle
obtuse angle
straight angle
Herman drew a 4 sided figure which had only one pair of parallel sides. What could this figure be?
Trapezium
Parallelogram
Square
Rectangle
Which quadrilateral has: All sides equal, and opposite angles equal?
Trapezium
Rhombus
Kite
Rectangle
Karen says every equilateral triangle is acute. Is this true?
Find the number of lines of symmetry of a circle.
A. 0
B. 4
C. 2
D. Infinite
The endpoints of a diameter of a circle are located at (5,9) and (11, 17). What is the equation of the circle?
What is the number of lines of symmetry in a scalene triangle?
A. 0
B. 1
C. 2
D. 3
How many diagonals does a rectangle has?
A quadrilateral whose diagonals are unequal, perpendicular and bisect each other is called a.
A. rhombus
B. trapezium
C. parallelogram

Didn't find what you were looking for?