Mathematics 2216
Homework 7 Solution
1. Prove that for any integers a, b, m, if a and m are relatively prime, b and m are relatively
prime, then ab and m are also relatively prime. (Also, typset the answer to this question
A
using L TEX) (Hint: Multiply t

MATH 2216 HW#5 Solution
Q1.
Q2. Given that a | b and a | c . We have
b = ak1 and c = ak2 for some k1 , k2 ! .
i) We have e = b + c = ak1 + ak2 = a(k1 + k2 ) .
Since k1 , k2 ! implies that k1 + k2 ! , we can conclude that a | e .
ii) We have
b+ f = c f = c

MATH 2216
HW # 1 Solution
1. Open the le sample.tex inside Canvas Typeset it to create the sample.pdf. Print it out
and attach it to the solutions of the next two problems.
2. Do Exercise 1.3 on p.1 of the notes
Answer: Let n be an odd natural number. We

MATH 2216
HW 3 Solution
Use mathematical induction to solve the following problems.
1. Let n be a positive integer. Show that the Fibonacci numbers, fi , satisfy
f1 + f2 + + fn = fn+2 1.
A
(Also, typset the answer to this question using L TEX)
Answer: We

MATH2216.02
Homework 2
Due on Friday 9/11
1. Prove directly that the sum of an even number and an odd number is always odd. (Also,
A
typset the answer to this question using L TEX)
Answer: Let a be an odd number and b an even number. By the denitions, we

Mathematics 2216
Homework 10 Solution
1. Given sets A and B dene the symmetric dierence
A B = (A B)
(A B).
(1) Prove that A B = (A B c ) (B Ac ).
(2) Use the result from part 1 to prove that A B = B A.
Answer: 1)
A B = (A B)
(A B)
(from denition)
c
(from

MATH 2216
HW 4 Solution
1. Let n be a positive integer. Using lHpitals rule and induction, prove that
o
xn
lim x = 0.
x e
Answer: When n = 1, we have by lHpitals rule that
o
x
1
lim x = lim x = 0
x e
x e
k
x
xk+1
We assume that lim x = 0 and want to prove

Mathematics 2216
Homework 8
Due Monday 10/26
1. Suppose that m is an integer such that 18 | m3 . Show that 36 | m2 . (Also, typset the
A
answer to this question using L TEX)
Answer: Since 3 | 18 and 18 | m3 , we have 3 | m3 . By the Euclid Lemma, this imp

MATH2216
Homework 6
Due on Friday 10/9, but Q1 is due by 8am of Wednesday 10/7 (through email)
1. Let m, n be positive integers with d as their greatest common divisor. Use Proposition 3.2
to show that
m m
gcd( , ) = 1.
d d
A
(Also, typset the answer to t

MATH2216
Homework 9 Solution
1. i) Prove that if A B and C D, then we have A C B D.
ii) Prove that A B = B if and only if B A.
Answer: i)
x A C x A and x C
x Band x C
(because A B)
x B and x D
xBD
(because C D)
Therefore we have A C B D.
ii) Assume that

Mathematics 2216
Homework 7
Due Oct 16 (Friday)
Name: n/A
1. Prove that for any integers a, b, m, if a and m are relatively prime, b and m are relatively
prime, then ab and m are also relatively prime. (Also, typset the answer to this question
A
using L T

MATH2216
Homework 6
Due on Friday 10/9, but Q1 is due by 8am of Wednesday 10/7 (through email)
1. Let m, n be positive integers with d as their greatest common divisor. Use Proposition 3.2
to show that
m n
gcd( , ) = 1.
d d
A
(Also, typset the answer to t

MATH 2216
HW # 5
Due Friday 10/2
1. Let m be a xed positive integer. Use mathematical induction to show that
n
j=m
j
n+1
=
, for all integers n m.
m
m+1
Answer:
Let P (n) :
n
j=m
ditions. P (1) :
that
1
1
=
2
2
j
n+1
= m+1 , for all integers n m. We rst c

MATH 2216
HW 3
Due Friday 9/18 (except question 1 which is due by Monday 9/14 7am)
Use mathematical induction to solve the following problems.
1. Let n be a positive integer. Show that the Fibonacci numbers, fi , satisfy
f1 + f2 + + fn = fn+2 1.
A
(Also,

MATH2216.02
Homework 2
Due on Friday 9/11
1. Prove directly that the sum of an even number and an odd number is always odd. (Also,
A
typset the answer to this question using L TEX)
Answer:
Let n be an even natural number, we know that n = 2k, where k is a