[Number Theory] How do you solve question 1? thanks 1. (i) Find all solutions to the linear Diophantine equation24 = 1260x +978y.(ii) Find all solutions to the linear Diophantine equation32 = 1260x +978y.2. Suppose that a, b and c are non-zero integers such that c | ab. Prove thatc| ged(a, c) gcd (b, c).3. Find all pairs of positive integers a, b such thate lcm(a, b) = 1061775 and ged(a, b) = 165?4. Without using a calculator, find the prime factorization of n = 12345654321 as follows.Hint: Note the following pattern: 11² = 121, 111² = 12321: Indeed, it is111 * 100+ 111 * 10 + 111 * 1 = 11100 +1110 + 111 = 12321.What is 1111²?Show that n is a perfect square, n = m². What is m?(i) Show that 3 m. What is m/3? (Compute by hand using long division.)(ii) Consulting the table of primes on p.249, as needed, a prime factor p of m/3 is obvious.What is p?(iii) Find the prime factorization of m/(3p) and hence the full factorization of n.(!)5. Satellite A flies in a circular orbit and passes over Toronto once every 36 hours. SatelliteB flies in another orbit and passes over Toronto every 15 hours. Satellite C passes overToronto every 22 hours. If they all passed over Toronto simultaneously at the start of classthis Tuesday, 3pm, September 19) When will be the next simultaneous passover?

Given information:Linear Diophantine Equations: To find:All solutions to the linear Diophantine…

Answered: s to the linear Diophantine equation 24…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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[Number Theory] How do you solve question 1? thanks

1. (i) Find all solutions to the linear Diophantine equation
24 = 1260x +978y.
(ii) Find all solutions to the linear Diophantine equation
32 = 1260x +978y.
2. Suppose that a, b and c are non-zero integers such that c | ab. Prove that
c| ged(a, c) gcd (b, c).
3. Find all pairs of positive integers a, b such thate lcm(a, b) = 1061775 and ged(a, b) = 165?
4. Without using a calculator, find the prime factorization of n = 12345654321 as follows.
Hint: Note the following pattern: 11² = 121, 111² = 12321: Indeed, it is
111 * 100+ 111 * 10 + 111 * 1 = 11100 +1110 + 111 = 12321.
What is 1111²?
Show that n is a perfect square, n = m². What is m?
(i) Show that 3 m. What is m/3? (Compute by hand using long division.)
(ii) Consulting the table of primes on p.249, as needed, a prime factor p of m/3 is obvious.
What is p?
(iii) Find the prime factorization of m/(3p) and hence the full factorization of n.
(!)
5. Satellite A flies in a circular orbit and passes over Toronto once every 36 hours. Satellite
B flies in another orbit and passes over Toronto every 15 hours. Satellite C passes over
Toronto every 22 hours. If they all passed over Toronto simultaneously at the start of class
this Tuesday, 3pm, September 19) When will be the next simultaneous passover?

Expert Solution

Step 1: Introduction.

Given information:

Linear Diophantine Equations:
$1260 x space plus space 978 y space equals space 24 1260 x space plus space 978 y space equals space 32$

To find:

All solutions to the linear Diophantine equation.

Concept used:

Linear Diophantine equations have solutions if and only if the greatest common divisor (GCD) of the coefficients of x and y divides the constant term (in this case, $24$ ).

Formula used:

Bézout's Identity: If a and b are integers, then there exist integers x and y such that $a x space plus space b y space equals space g c d left parenthesis a comma space b right parenthesis$ .