21-127 Concepts of Mathematics
Due Thursday, February 2
Hand in complete solutions to all required problems (5 points each) at the beginning
You are not required to do or hand in bonus problems, though you
should read and try them.
You will receive a small bonus (one point each) for up
to two complete and correct bonus problems. Papers typeset using L
X will also
receive one bonus point.
be an integer. Prove that
leaves a remainder of 0, 1 or 4 when divided
2. This problem concerns lattice points in 3-dimensional space.
be the largest possible number of lattice points so that no pair of the
points contains a lattice point on the segment between them. Find
give an example of
(b) Prove that in any set of
+ 1 points that there must be some pair where
the segment between them contains another lattice point.
(c) Optional: Generalize this problem to
be any real number such that