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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).

Advanced Engineering Mathematics
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 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?
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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