1260x + 9789. 2. Suppose that a, b and c are non-zero integers such that c ab. Prove that c| ged(a, c) gcd (b, c).
1260x + 9789. 2. Suppose that a, b and c are non-zero integers such that c ab. Prove that c| ged(a, c) gcd (b, c).
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 35E
Related questions
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[Number Theory] How do you solve question 2? 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?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5282306a-36ce-4fa5-aa61-049a72c51220%2Fa81b5ad1-9ca3-479a-866b-f23bda98ccd9%2Fqtsv9e_processed.png&w=3840&q=75)
Transcribed Image Text: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?
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