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?
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?
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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[Number Theory] How do you solve question 3? thanks
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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