Math 350 - Fall 2002 - Number Theory

1. (Schumer 2.1 #2)

a) Use the Euclidean Algorithm to find gcd(462,2002)

b) Find x and y such that 462*x+2002*y=gcd(462,2002)

2. (Schumer 2.1 # 6) Show that any two consecutive squares are relatively prime

3. (Schumer 2.1 #18) Determine whether the following linear Diophantine equations are solvable. If so. find all solutions.

(a) 14*x+35*y=106

(b) 51*x-153*y=34

(d) 5*x+55*y=125

4. (Schumer 2.2 #9)

(a) What number between 1 and 1000 is congruent to 2 (mod 7), 3 (mod 11), and 8 (mod 13) ?

(b) What number between 1 and 1000 is congruent to 14 (mod 7), 33 (mod 11), and 28 (mod 13) ?

5. Suppose that a and b are relatively prime, and that ab is a perfect square. Show that a and b are perfect squares. <\p>

Bonus: Show that among any ten consecutive integers, there is at least one that is relatively prime to the other nine. <\p>