Polyas Method, Mathematical Systems, and basic Cryptology

A situation that confronts the learner or a person, that requires resolution, and for which the path to the solution/answer is not immediately known.

Problem

It is a mathematical task where we know what is ask but do not know the direct way of doing it.

Problem

Successful problem solvers use a variety of techniques when they attempt to solve a problem. What step is this?

Step 2: Devise a plan

What are the 11 ways to devise a plan?

1. Working backwards

2. Logical reasoning

3. Finding a pattern

4. Creating a visual representation

5. Write an equation

6. Solve a similar but simpler problem

7. Organizing data

8. Adopt a different point of view

9. Considering extreme cases

10. Accounting for all possibilities

11. Intelligent guessing and testing

What are the 4 steps of POLYAS method?

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Review the solution

True or false: Learning and applying the most elegant method might be a goal to keep in mind.

True

The Swiss mathematician who pioneered the modern approach to congruence in 1750. he explicitly introduced the idea of congruence modulo a number N.

Leonhard Euler

Modular arithmetic was further advanced by ___ in his book published in 1801.

Carl Friedrich Gauss

What step does this question involve?

• What is known & unknown?

• How do you relate them?

• What is given?

Understand the problem

What step does this question involve?

• Does it make sense?

• Is it reasonable?

• Will it apply to offer problems?

Reexamine the solution

The general outline which requires checking and performing calculations.

Cary out a plan

True or false: problem solving skill is transferrable skill.

True

In Mathematics, modular arithmetic is sometimes called ___

Clock arithmetic

Is a system of arithmentic for integers, where numbers wrap around after they reach a certain value —-the modulus.

Modular arithmetic

has a 12-hour deisgn

clock

designated whether it is before noon or after noon

time

n its most elementary form, arithmetic is done with a count that resets itself to zero every time a certain whole number N greater than one, known as the modulus (mod), has been reached.

modular arithmetic

the statement a = b (mod n) is called ____

congruence

true or false: The notion of modular arithmetic is related to that of the remainder in division.

true

he operation of finding the remainder is sometimes referred to as the ___

modulo operation

Tell whether the congurence is true or not:

• 29 = 8 mod 3

• 7 = 12 mod 5

• 15 = 4 mod 6

• true

• true

• not

evaluate (23 + 38) mod 12

1 is the remainder

Evaluate (33-16) mod 6

5

Evaluate (15 x 23) mod 11

4

What are the three application of modular arithmetic?

ISBN, UPC, Credit cards

This 13-digit number was created to help ensure that orders for books are filled accurately and that books are catalogued correctly.

ISBN

The 3th digit in the ISBN is

Check digit

What is the meaning of UPC?

Universal Product Code

Another coding scheme that is closely related to ISBN which is placed on many items and is particularly useful in grocery stores.

UPC

This gives the store manager accurate information about inventory and the buying habits of the store’s customers.

UPC

Is a 12-digit number that satisfies a congruence equation that is similar to the one for ISBNs.

UPC

The primary coding method i of credit card is based on the ___ which uses mod 10 arithmetic.

Luhn Algorithm

Credit cards are usually __ to digits long. The first one to four digits are used to identify _.

13, 16, card issuer

Is the discipline devoted to secrecy systems

Cryptology

Is a branch of cryptology that deals with the design and implementation of secrecy systems

cryptography

involved breaking these secrecy systems

cryptoanalysis

is a message that is to be altered into a secret form

plaintext

refers to the method for altering a plaintext message into a ciper text.

cipher

Changing the letter of the plaintext used a

transformation

determines a particular transformation from a set of possible transformations from a set of possible transformations

key

is the process of changing plaintext into cipher text

encryption or enciphering

is the reverse process of changing the cipher text back to the plaintext

decryption pr deciphering