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?