MATH 340
Assignment 1
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(1) Prove for all positive integers n,
nXi=1i3 =n(n+1)22:
(2) Find gcd(2424;772) and express it as a Z-linear combination of 2424 and
772.
(3) (a) Prove if d is a common divisor of a and b, then d divides gcd(a;b).
(b) Prove if m is a common multiple of a and b, then lcm(a;b) divides m.
(4) Suppose p is a positive prime integer and k is an integer satisfying 1 k
p 1. Prove that p dividespk.
(5) (a) Determine the elements in Z=15Z that have multiplicative inverses (and
nd the inverses).
(b) Give an example of an equation of the form. [a]X = [b], [a]6= [0], that
has no solution in Z=15Z.
(c) Give an example in Z=15Z where [a][b] = [a][c], but [b]6= [c].
(6) Show that if [a] has a multiplicative inverse in Z=mZ, then gcd(a;m) = 1.
(7) In the integers, the equation X2 = a has a solution if and only if a is a
positive perfect square or zero. For which [a] does the equation X2 = [a]
have a solution in Z=7Z (resp. Z=8Z, Z=9Z)?