MATH 327, HOMEWORK #4
Problem 1. Suppose nis a natural number, and let Adenote the set
of all divisors of n. Using the relation ”|” of divisibility we obtain a
partial order on A.
(a) Draw the associated Hasse diagram for the case n= 8.
(b) Draw the associated Hasse diagram for the case n= 27.
(c) Draw the associated Hasse diagram for the case n= 12.
Problem 2.
(a) Find d= GCD(2646,495) using the Euclidean algorithm.
(b) Find integers uand vsuch that
u·2646 + v·495 = d.
Problem 3.
(a) Show that if
6m4n= 6k4l
for natural numbers m, n, k, l then
m=kand n=l
(b) Find an example where
m6=kand n6=l
and yet
8m4n= 8k4l
Problem 4.
(a) For which natural numbers is 3aa perfect square?
(b) For which natural numbers is 9aa perfect square?
Problem 5.
(a) Find GCD(6,8) and GCD(36,48) using the Euclidean algormithm.
(b) Find GCD(6,8) and GCD(36,48) using prime factorization.
(c) Guess a formula GCD(a, b) and GCD(a2, b2).Give two examples
that agree with your formula.
(d) Prove your formula.
Problem 6.
(a) Find GCD(3,1033).
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