Mat 246F - Practice problems
Divisibility and modular arithmetic
1. Suppose that a, b and c are integers such that a2 + b2 = c2 . Prove that at least one of a, b and
c is divisible by 3.
2. Prove that 6 divides n(n2 + 11) for every integer n.
3. Prove tha
Mat 246 Practice Problems
Divisibility and primes, Fermats Little Theorem and Wilsons Theorem
1. If n is an integer and 7 | n, prove that exactly one of n3 + 1 and n3 1 is divisible by
7.
2. Let n be a natural number. Prove that n5 and n have the same las
Practice Final 1
1. (a) What is (20100 ) where is Eulers -function?
(b) Find an integer x such that 140x 133 (mod 301).
Hint: gcd(140, 301) = 7.
2. (a) Prove, by mathematical induction, that 1 + 2 + 3 + . + n =
natural number n.
n(n+1)
2
for every
(b) Pro
MAT 246 - Material covered in Sept. 10 class
Note: The numbering of lemmas and theorems is the same as in the textbook. Many of the
examples discussed in class are dierent from the examples in the textbook. The details
of the proofs and the examples are n
MAT 246H1F (2014) Problem Set 1
Hand in at the beginning of tutorial on Thursday Sept. 25. Late problem sets will not be
accepted.
NOTE:
Students are expected to write up problem sets independently. Please refer to the
course information sheet for inform
University of Toronto
Department of Mathematics
MAT246H1S
Concepts in Abstract Math
Midterm Examination
February 23, 2010
Examiners: F. Murnaghan, P. Rosenthal
Duration: 1 hour 50 minutes
Last Name:
Given Name:
Student Number:
Section: (Day or Evening)
No
Mat 246H1F - Term test
October 27, 2010
No calculators or other aids allowed
10 marks per question; 100 marks total
Duration: 2 hours
1. Find the remainder after dividing 6!(130)121 +7(1475)101 by 8. Please show your work.
2. Use mathematical induction to
Department of Mathematics
University of Toronto
MAT 246Y
Term Test #1
October 24, 2006
Examiners: P. Rosenthal
Duration: 1 1 hours
2
LAST NAME:
FIRST NAME:
STUDENT NUMBER:
There are ten questions, each of which is worth 10 marks.
This paper has a total of
Problem Set 4 Solutions
0.1
Problem 1
0.1.1
Let S be the set of all closed intervals in R with endpoints in Q( 5).
Claim : The cardinality of S is 0 ( ?)
We give a very hands on proof. Let Q+ denote the set of positive rationals. Consider the map : S
Q(
Problem Set 3 Solutions
0.1
Problem 1
0.1.1
4
Claim:
2 + 15/ 3 2 is not constructible.
Recall the following general facts: i) all naturals are constructible, ii) the constructible numbers are closed
under the operations: taking square roots, multiplicatio
Problem Set 1 Solutions by Parker Glynn-Adey.
Remarks that are only meant to help understand the solutions written below
are included in braces and italics. They are simply editorial remarks.
Marking Scheme:
Problem 1: 10pnts
Problem 2: 10pnts
Problem
MAT 246 Cardinality denitions and theorems
Note: These notes do not contain any proofs or explanations. For proofs, please refer to
the other notes on cardinality and to material covered in class.
We will use the notation |S | to denote the cardinality of
(1) Which of the following is a number eld?
(a) the set of all nonnegative rational numbers;
(b) the set of numbers of the form a + b 2 c 3
+
where a, b, c Q;
(c) the set of numbers of the form a + b 2 + c 4 2 + d 4 8 where
a, b, c, d Q.
Hint: Look at th
Hw7
Do the following problems from the text:
page 83: 3, 6, 7, 9,11, 12, 14
Reading: Chapters 9, 10
(1) #3 on page 83:
(a) nd i
(b) nd 15 8i.
Solution
(a) since | i| = 1 we write i = 1 (cos( ) + i sin( ). There2
2
1
1
fore, i = (cos( ) + i sin( ) = ( 2 i
(1) Let a, b be odd integers.
Prove that a2 + b2 is irrational.
Hint: Look at divisibility by the powers of 2.
(2) Prove that for any real numbers a < b there exists an irrational
number c such that a < c < b.
Hint: Look at the numbers of the form q 2 whe
Practice Final 3
1. The Fibonacci sequence is the sequence of numbers F (1), F (2), . . . dened by the
following recurrence relations:
F (1) = 1, F (2) = 1, F (n) = F (n 1) + F (n 2) for all n > 2.
For example, the rst few Fibonacci numbers are 1, 1, 2, 3
Practice Final 2
1. Using induction prove that
12 + 32 + . . . + (2n + 1)2 =
(n + 1)(2n + 1)(2n + 3)
3
2. Let a, b, c be natural numbers.
(a) Show that the equation ax + by = c has a solution if and only if (a, b)|c.
(b) Find all integer solutions of 6x +
Practice Final 4
1. Use induction to prove that
1 + 2q + 3q 2 + . . . + nq n1 =
for any real q = 1 and any natural n.
1 (n + 1)q n + nq n+1
(1 q)2
2. (a) Find 43! (mod 45).
(b) Find the last digit of 32014 .
3. Let p1 , p2 be distinct prime numbers.
Prove
(1) Prove that
2
3
n
n+2
1
+ 2 + 3 + . + n = 2 n
2 2
2
2
2
(2) Prove that
1 + 2q + 3q 2 + . . . + nq n1 =
1 (n + 1)q n + nq n+1
(1 q)2
(3) The Fibonacci sequence is the sequence of numbers F (0), F (1), . . .
dened by the following recurrence relations:
F
(1) Give a proof by induction of the following statement used class:
Let m 1 be a natural number. Then for any n 0 there exists an
integer r such that 0 r m and n r pmod mq.
100
(2) (a) Find 23 p mod 5q
100
(b) Find the last digit of 23 .
Hint: use part a
(1) Using the method from class write a table of all prime numbers
100. Explain why you only need to cross out the numbers divisible
by 2, 3, 5 and 7.
(2) Let p1 , p2 be distinct primes. Using the Fundamental Theorem of
Arithmetic prove that a natural nu
MAT 246S
Solutions to the Term Test
Winter 2013
(1) (10 pts) The pigeonhole principle states that if n items are put into m pigeonholes
with n > m, then at least one pigeonhole must contain more than one item.
Prove the pigeonhole principle by induction i
(1) Using the Euclidean Algorithm prove that if gcda, bq 1 and ac, bc
then abc.
(2) Using the Euclidean Algorithm nd gcd291, 573q and integer x, y such
that 291x 573y gcd291, 573q.
(3) (a) Find all integer solutions of the equation
25x 10y 200
(b) Find al
(1) Let p1 , p2 be distinct prime numbers.
Using the method from class give a careful proof of the formula
(2)
(3)
(4)
(5)
k k
k
k
k
p11 p22 q p11 p11 1 q pk2 p22 1 q
2
Let a, b, c be natural numbers. Let a, b, cq be the largest natural number
that divid
MAT 246S
Practice Term Test 2
Winter 2014
(1) Find the formula for the sum 1 2 2 3 + 3 4 . . . + (2n 1) (2n) (2n) (2n + 1)
and prove it by mathematical induction.
(2) Find the remainder when 6100 is divided by 28.
(3) Find the integer a, 0 a < 37 such tha
MAT 246S
Practice Term Test 3
Winter 2014
(1) Find a mistake in the following proof.
Claim: 1 + 2 + . . . + n = 1 (n + 1 )2 for any natural n.
2
2
We proceed by induction on n.
a) The claim is true for n = 1.
b) Suppose we have already proved the claim fo
(1) Let S = P (N)
Show that |R| |S|.
Hint: Since |R| = |(0, 1)| its enough to show |(0, 1)| |S|. Take
a number x (0, 1), look at its decimal expression x = 0.a1 a2 a3 . . .
and take a subset of N given by numbers whose decimal expressions
are 1a1 , 1a1 a2
MAT 246S
Practice Term Test 1
Winter, 2014
Prove by mathematical induction that n3 + 5n is divisible by 6 for any natural n.
Find the remainder when 7101 is divided by 101.
Find the integer a, 0 a 20 such that 13a 1(mod 20).
Prove that if m 1(mod (n) and