Homework 1
MAT3166 Fall 2015
Due September 30 in class.
(1) For which x R do we have [2x] = 2[x]?
(2) Show that for any x R, [x] + [x + 1 ] = [2x].
2
(3) Evaluate
n
1
k=2 k2 1
(4) Evaluate
n
k=1 (1
. Hint:
1
k2 1
1
= 1 ( k1
2
1
)
k+1
.
1/k 2 ).
(5) Show

Homework 3
MAT3166 Fall 2015
Due November 2 in class. The rst 6 questions concern material covered in the midterm.
(1) Find all solutions of the system of congruences
x 7 mod 3
x 8 mod 10
x 0 mod 7.
(2) Find all integers of the form 7n + 1 whose last digi

Practice problems for midterm
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
MAT3166
Show that the harmonic numbers satisfy H2n 1 + n/2.
Show that for n 1, (2n)! < 22n (n!)2 .
Find a simple formula for f1 +

Homework 2
MAT3166 Fall 2015
Due October 14 in class.
(1) Show that there are arbitrarily large gaps between primes. Hint: consider n! + 2, n! +
3, ., n! + n.
(2) Determine whether there are innitely many triples of primes of the form (n, n +
2, n + 4).
(