Math 110 - Final Practice Solutions
Intro to Math, Winter 2013
1. The goal of this problem is to nd 7/30 (mod 43).
(a) Find a solution to 30x + 43y = 1 with x positive.
(b) Use this to nd 1/30 (mod 43).
(c) Use this to nd 7/30 (mod 43), which should be a
Syllabus and practice problems for Midterm 2
1
Solving equation ax + by = d
See Ch 9. p 107-108 and Ex. 9.4.1-9.4.4. The answers are posted on Canvas in the folder
Assignments.
2
2.1
Primes; prime factorization
Some theory
A number p > 1 is called prime i
MATH110
Quiz 1
Note: You do not need to compute the numerical values.
1a) How many dierent 8 letter words can one write using letters H
and T?
b) How many dierent words can one write using 3 letters H and 5
letters T?
c) Tossing the coin 8 times, what are
Guidelines for MATH 110
1
General Information
Lecture: MWF 3:00-3:50pm LUNT 103
Discussion sessions:Th: 3:00-3:50 103
Instructor: Dmitry Tamarkin
Oce: Lunt B6; oce hours MWF 4-5pm
e-mail [email protected]
text: B. Gross, J. Harris, The
Syllabus and practice problems for Midterm 2
1
Solving equation ax + by = d
See Ch 9. p 107-108 and Ex. 9.4.1-9.4.4. The answers are posted on Canvas in the folder
Assignments.
2
2.1
Primes; prime factorization
Some theory
A number p > 1 is called prime i
Eulers theorem; roots
mod N
1. Eulers theorem
Let N and a be numbers; we assume gcd(a, N ) = 1 i.e. a and N are
relatively prime. Eulers theorem then says that
a(N ) 1(
mod N ),
where (N ) is the so called Eulers function: (N ) is equal to the
number of n
Answers to the exercises from Ch. 10
10.2.1
1. 85 = 5 17;
2. 342 = 2 3 3 19;
3. 851 = 23 37;
4. 137 = 137.
10.3.1 420 < 21, therefore, it suces to sieve out all multiples of
2,3,5,7,11,13,17, and 19:
2, 5 3 2
2, 3 13 2 3, 5 2, 7 11 2, 3, 17
B
B 412
B
B
B
Answers to the exercises from Ch. 15,16,17
15.2.1 3
15.3.3 cubes: 0,1,3,2,4; 4-th powers 0,1.
15.4.1 1. 8, 2. 10; 12; 3. 21;
15.4.2 1. 4, 2. 5; 3. 2; 4. 3 or 4, 5. mod 5.
15.4.3 mod 10: 0,9,8,7,6,5,4,3,2,1; mod 11: 0,10,9,8,7,6,5,4,3,2,1. The
last line mo
Answers to the exercises from Ch.8 and 9
8.2.1 1. 10, 2. 21, 3. 45, 4. 42, 5. 132, 6. 216;
8.4.2 1. 3; 2. 1; 3. 54; 4. 1;
8.4.3 1. 195; 2. 637; 3. 432;
8.4.4 Number of numbers from 600 to 6000 is 5401; number of numbers
from 600 to 6000 divisible by 39 is
MATH110
Syllabus and practice problems for the I-st midterm
1. Introduction
The exam will be on chapters 1,2,3,4,5,8, and 9. Below you will nd
all types of problems that we have studied and you may expect similar
problems on the exam. There are solutions
Math 110: Quiz 6 Solutions
Intro to Math, Winter 2013
(1) Compute 2/13 (mod 17) by rst nding a solution in whole numbers to 13x + 17y = 2.
(2) Compute 221 (mod 18) by writing 21 as a sum of powers of 2 and using a table giving the
values of x, x2 , x4 , x
Math 110: Quiz 5 Solutions
Intro to Math, Winter 2013
(1) Compute (360). (Remember that this is the number of things smaller than 360 which are
relatively prime to 360.)
(2) Compute the following three expressions in arithmetic mod 10.
1, 349 + 2, 342 (mo
Math 110 - Midterm 1 Practice Solutions
Intro to Math, Winter 2013
1. How many numbers between 474 and 3867 are divisible by 7? How many are divisible by 14?
Solution. The rst number in the given range divisible by 7 is 476 which is 7 68. The last number
Math 110: Midterm 1 Solutions
Intro to Math, Winter 2013
1. Consider phone numbers consisting of a 3 digit number between 130 and 895 inclusive followed
by a 4 digit number between 1200 and 6780 inclusive. How many such phone numbers are there if
the 3 di
Math 110 - Midterm 2 Practice Solutions
Intro to Math, Winter 2013
1. Find the greatest common divisor and least common multiple of 60760 and 48763 using the
Euclidean algorithm.
Solution. The Euclidean algorithm gives:
60760 = 1 48763 + 11997
48763 = 4 1
Math 110: Midterm 2 Solutions
Intro to Math, Winter 2013
1. (10 points) Consider the numbers 246 and 180.
(a) Find the greatest common divisor of these using the Euclidean algorithm.
(b) How many numbers between 14, 000 and 40, 000 are divisible by both 2
Math 110: Quiz 1 Solutions
Intro to Math, Winter 2013
(1) Consider phone numbers consisting of a 3 digit number followed by a 4 digit number. How
many such phone numbers are there if the 4 digit number portion must be between 1234 and 7912
inclusive? Note
Math 110: Quiz 2 Solutions
Intro to Math, Winter 2013
(1) How many numbers between 27 and 540 inclusive are divisible by neither 3 nor 5. Use the
fact that the numbers which are divisible by both 3 and 5 are the same those which are divisible
by 15. You d
Math 110: Quiz 3 Solutions
Intro to Math, Winter 2013
(1) Use the Euclidean algorithm to nd the greatest common divisor and least common multiple
of 14035 and 3570.
(2) Find a solution in whole numbers to 16x + 70y = 4.
Solution. (1) The Euclidean algorit
Math 110: Quiz 4 Solutions
Intro to Math, Winter 2013
(1) In a crazy new dance sweeping the nation, called The Mathematician: like Gangnam Style,
but better, you can either move 30 steps forward or 23 steps backward any number of times. Using
only these t
Answers to the exersises from Ch12 and Ch5
12.2.1
1. Yes, n/m = 2 7 13;
2. No.
3. No.
4. Yes, n/m = 2 7 11.
12.3.2
1.104 = 23 131 . The number of divisors is 4 2 = 8;
2. 210 = 2 3 5 7. the number of divisors is 16.
3. 384 = 27 31 , the number of divisors