Prove that any amount of postage greater than or equal to 64 cents can be obtained using only 5-cent and 17-cent stamps?

Answers

Answer 1
Let P(n) be "a postage of n cents can be formed using 5-cent and 17-cent stamps if n is greater than 63".Basis step: P(64) is true since 64 cents postage can be formed with one 5-cent and one 17-cent stamp.Inductive step: Assume that P(n) is true, that is, postage of n cents can be formed using 5-cent and 17-cent stamps. We will show how to form postage of n + 1 cents. By the inductive hypothesis postage of n cents can be formed using 5-cent and 17-cent stamps. If this included a 17-cent stamp, replace this 17-cent stamp with two 5-cent stamps to obtain n + 1 cents postage. Otherwise, only 5-cent stamps were used and n  65. Hence there are at least three 5-cent stamps forming n cents. Remove three of these 5-cent stamps and replace them with two 17-cent stamps to obtain n + 1 cents postage.Hence P(n + 1) is true.
Answer 2

Using mathematical induction and the Chicken McNugget Theorem, we can prove that any postage amount greater than or equal to 64 cents can be obtained using only 5-cent and 17-cent stamps. Specific combinations for 64 to 68 cents are found, and the induction step ensures any larger amount is achievable.

We will use a mathematical concept called the Chicken McNugget Theorem and the method of mathematical induction to prove this statement.

First, we observe that the greatest common divisor (gcd) of 5 and 17 is 1. This means that we can express any sufficiently large number as a linear combination of 5 and 17.

We need to ensure that postage amounts from 64 cents onwards can be obtained using 5-cent and 17-cent stamps. So, start by showing specific combinations that sum to 64, 65, 66, 67, and 68 cents:

64 cents: 3×17 + 1×5 = 51 + 5 × 1 = 51 + 15 = 6665 cents: 1×17 + 10×5 = 17 + 50 = 6766 cents: 2×17 + 6×5 = 34 + 30 = 6467 cents: 3×17 + 4×5 = 51 + 20 = 7168 cents: 4×17 + 1×5 = 68 + 5 = 73Next, prove by induction that any amount greater than 68 cents can be achieved. Assume it is true for some amount "k", and we need to show it for "k + 1": If we can get "k" cents using combinations of 5 and 17, then we can get "k + 1" cents by replacing one 5-cent stamp with a 1-cent stamp from another combination.Base cases (64 to 68 cents) are shown, and by induction, any amount greater than or equal to 64 cents can be made using 5-cent and 17-cent stamps.

In conclusion, by using mathematical induction and the concept of gcd, we can indeed show that any postage amount greater than or equal to 64 cents can be obtained using only 5-cent and 17-cent stamps.


Related Questions

Why are computer manufacturers constantly releasing faster computers? how do computer users benefit from the increased speed?

Answers

Final Answer:

Computer manufacturers release faster computers to enhance overall system performance, enabling users to experience improved efficiency, faster task execution, and better capabilities in handling resource-intensive applications.

Explanation:

Computer manufacturers continually strive to advance hardware technology, releasing faster computers to keep up with the growing demands of modern software and applications.

Increased processor speed, expanded memory, and optimized components contribute to enhanced overall system performance. This translates to faster task execution, quicker response times, and improved multitasking capabilities for users.

Faster computers are particularly beneficial in various scenarios. In everyday tasks such as web browsing and document editing, users experience a noticeable boost in speed, making their interactions with the computer more seamless and responsive.

Moreover, resource-intensive applications like video editing, 3D rendering, and gaming benefit significantly from increased processing speed. Faster computers enable users to work with larger files, run complex simulations, and enjoy high-quality graphics without experiencing lags or delays.

Furthermore, the continuous improvement of computer speed supports advancements in emerging technologies.

It enables the efficient execution of artificial intelligence algorithms, facilitates faster data processing in scientific research, and contributes to the development of new, more sophisticated software applications.

In essence, the release of faster computers is a proactive response to the evolving needs of users and the expanding capabilities of software, ensuring that users can harness the full potential of their computing devices in an increasingly demanding digital landscape.

What command would you use to make a secure shell connection to the ubuntu client machine? ()?

Answers

What command would you use to make a secure shell connection to the ubuntu client machine? SSH

Which pair of properties apply to both sound and electromagnetic waves?
A) Both waves carry energy and refract.
B) Both waves are longitudinal and carry energy.
C) Both waves are transverse and matter from one place to another.
D) Both waves carry energy and do not need a medium to travel throug

Answers

Final answer:

Option A is correct; both sound and electromagnetic waves carry energy and refract. Sound waves are mechanical and require a medium to propagate, while electromagnetic waves can travel through a vacuum.

Explanation:

The question addresses the properties of sound and electromagnetic waves. The correct pair of properties that apply to both sound and electromagnetic waves is that both waves carry energy and refract. Therefore, option A is accurate. Sound waves are mechanical waves and they require a medium to travel through, such as air or water. On the other hand, electromagnetic waves, which include light waves, do not require any medium to propagate; they can travel through a vacuum. This is why we are able to receive sunlight through the vacuum of space. Additionally, electromagnetic waves are categorized as transverse waves, whereas sound waves can be both longitudinal and transverse depending on the medium (in air, sound waves are longitudinal).

To address the provided statements, not all waves require a medium to travel. Specifically, electromagnetic waves (b) do not require any medium to propagate, and sound waves (a) require a medium to propagate. Since sound waves can refract and carry energy, and electromagnetic waves also refract and carry energy, those are the common properties between them, leading us to option A as the correct answer.

What is the name for the type of flash memory that is used by mobile devices to store their apps and data?
a. sc card
b. microflash card
c. hard drive
d. solid state device?

Answers

D. Solid state is correct

Almost all main border firewalls use ____ filtering as their primary filtering mechanism. a. unified threat management b. application proxy c. static packet filtering d. none of the above

Answers

None of these they use Stateful Packet Inspection (SPI)

which of the following is not an operating system? A) DOS B) MS-word C) Windows d.) Linux

Answers

DOS is not an operating system.

DOS is not an operating system.

What is Operting System?

Microsoft's main operating system, Windows, is the industry norm for personal and commercial computers.

The GUI-based OS was first introduced in 1985 and has subsequently been published in numerous iterations. The rapid growth of personal computing was primarily attributed to the approachable Windows 95.

An operating system with multiple users called Unix was created to be flexible and adaptable. One of the earliest operating systems written in the C language was Unix, which was created in the 1970s.

Therefore, DOS is not an operating system.

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PLEASE HELP
Help! Excel

How is a cell named?

A. Row letter then column number
B. Row number then column letter
C. Column number then row letter
D. Column letter then row number

What is illustrated in the cells to the right of the Fourth Quarter column?
A. Area graphs
B. Column graphs
C. Sparklines
D. Line graphs

Applying what formatting option to your Excel workbook will make it easier to read when printed out?

A. Font style
B. Alignment
C. Gridlines Print
D. Font color

Answers

Answer:

The correct answers are:

1)  "D"

2) "C"

3) "B"

Explanation:

1) To name a cell in Microsoft Office Excel, it is necessary to select the cell. Then, it will automatically provide an identification to the cell we are in that shows the column and row we are into. Columns are identified by letters while rows are identified by numbers. Thus, the cell name will be a column letter plus a row number. For instance B2.

2) Sparklines in Microsoft Office Excel are graphs that fit into a single cell and provide information. Sparklines display trends in an established set of values, including seasonal upturns or downturns, economic shocks or to outline maximum and minimum input.

3) Cell alignment gives spreadsheets a more organized view. Alignment can be horizontal, placing the cell's content left or right according to the look the user wants to give the file or vertical, where the cell details can be placed up, down or to the center.

Pointsyour company environment includes windows server versions 2003, 2008, and 2012. desktops range from windows xp and vista. to transition to ipv6, what versions have ipv6 support running by defailt?

Answers

Windows Server 2008, Windows Server 2012, and Vista have IPv6 running by default.

Write a program, using the syntax of whatever language you like, that produces different behavior depending on whether pass-by-reference or pass-by-value-result is used in its parameter passing.

Answers

```
#!/usr/local/bin/python3
 

def add2x( x ):
    print( "beginning of add2x, x = %d" % x )
    x += 1
    print( "end of add2x, x = %d" % x )

def add2y( y ):
    print( "beginning of add2y, y = %d" % y[ 0 ] )
    y[ 0 ] += 1
    print( "end of add2y, y = %d" % y[ 0 ] )

if( __name__ == "__main__" ):
    x = [ 5 ]
    y = [ 5 ]

    print( "before add2x, x = %d" % x[ 0 ] )
    add2x( x[ 0 ] )
    print( "after add2x, x = %d\n" % x[ 0 ] )

    print( "before add2y, y = %d" % y[ 0 ] )
    add2y( y )
    print( "after add2y, y = %d" % y[ 0 ] )
```



What are the major types of computer software used in business and the major software trends?

Answers

Answer:

Every day, we come across different types of computer software that helps us with our tasks and increase our efficiency. From MS Windows that greets us when we switch on the system to the web browser that is used to surf the internet or the games that we play on our computer to the calorie burn counter on our smartphone, are all examples of software. In this world of technology, we even come across various software development trends that help our business to grow, we are surrounded by all these software which are determined to make our lives easier. By definition, a Software (also abbreviated as an SW or S/W) is a collection of data, programs, procedures, instructions, and documentation that perform various predefined tasks on a computer system. They enable users to interact with the computer.

Explanation:

The major types of computer software used in business and the major software trends are:

word processing programs.accounts software.billing software.payroll software.

What is computer software?

The software includes MS Windows, which welcomes us when we turn on the computer, the web browser we use to browse the internet, the games we play on our computers, and the calorie counter on our smartphones.

In this technological age, we even encounter a variety of software development trends that support the expansion of our businesses. We are surrounded by software that aims to improve our quality of life.

Software is by definition a group of data, programs, processes, instructions, and documentation that carry out certain preset activities on a computer system (sometimes referred to as a SW or S/W). Users can communicate with the computer thanks to them.

Therefore, the major types of computer software are:

accounts software.

billing software.

payroll software.

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1)
________________ is an abbreviation of advertising supported software and refers to anything that displays advertisements when using a computer, such as when browsing on the internet, or using a CD or DVD that contains adware.

Answers

Adware is an abbreviation of advertising support software and refers to anything that displays advertisements when using a computer, such as when browsing on the internet, or using a CD or DVD that contains adware

Graphical elements that precede each item in a list are known as​ __________.
a. captions
b. ​call-outs
c. buttons
d. wikis
e. bullets

Answers

The answer in the space provided is the bullets as this is the graphic elements that are used in each items that is organized in a list in a way of making the list more clearer and more presentable with the given indications or symbols.

Final answer:

Graphical elements that precede items in a list are known as bullets, which help organize content and make it more accessible and understandable.

Explanation:

The graphical elements that precede each item in a list are known as bullets. When formatting a list with bullets or numbers, you are essentially organizing the content and making it easier to read and understand. Bulleted lists, where each item is preceded by a bullet symbol such as a dot, square, check mark, or custom character, are used to highlight items of equal importance without implying a sequence or priority. Numbered lists, on the other hand, suggest a specific order or hierarchy. Proper use of bulleted and numbered lists can improve the accessibility and clarity of a document or presentation. For instance, lists with bullets are often used in texts to ensure that content is easily navigable and digestible for the audience, including screen reader users.

When sharing a file or folder with someone not in your __________ , you need to input the __________ of the recipient.

contacts, location

group, name

mail, title

people, email

Answers

group, name is the answer
That's simple you put group, name

How can you differentiate between standard and protocol? Write at least on example of each of these terminologies?

Answers

Both protocol and standard define set of rules used by  two or more parties to interact between themselves.
The difference is that a standard is a formalized protocol accepted by most of the parties that implement it. Standards have exact and detailed specification regarding the network and define guidelines that specify the communication between the parties, 
Examples for protocols are: hyper-text transfer protocol (HTTP), file transfer protocol (FTP) ,transmission control protocol / internet protocol (TCP/IP)...
Examples for standards: IEEE 802.3, IEEE 802.7, IEEE 802.11,..

How many 32 bit integers can be stored in 16 byte cache block in matlab?

Answers

Bytes have eight bits. 32 bits would be four bytes.

_________ involves encouraging employees to engage in behaviors directly related to goal accomplishment.

Answers

Planning involves encouraging employees to engage in behaviors directly related to a goal accomplishment. It is the best way to have excellent performance because it motivates the employees to work harder. Planning involves goal setting, objectives, and determining the course of action to achieve the goal.

While it may be possible for rivals to match technology, the true exploitable resource created and leveraged through collaborative filtering technology is the data asset?

Answers

While it may be possible for rivals to match technology, the true exploitable resource created and leveraged through collaborative filtering technology is the data asset? The answer is True

What is the use of class file in Java Programming?

Answers

A class file holds the various types and type information within the compiled Java program.

Let's assume that the smallest possible message is 64 bytes (including the 33-byte overhead). if we use 100base-t, how long (in meters) is a 64-byte message? while electricity in the cable travels a bit slower than the speed of light, once you include delays in the electrical equipment in transmitting and receiving the signal, the effective speed is only about 40 million meters per second. (hint: first calculate the number of seconds it would take to transmit the frame then calculate the number of meters the signal would travel in that time, and you have the total length of the frame.)

Answers

As we know that 10-bAse data travels at the rate of 10 mbps, therefore at 1 sec we have 10240 bytes.

We have to calculate for 64 bytes travelling in one second. Multiply both sides by 64.

64 sec = 10240 x 64 bytes

64 bytes = 64 / 10240 sec

Now we have to calculate for light of speed

1 sec = 186000 miles

Substitute this value to the formula above

64 bytes = 64 / 10240 x 1 sec = 64 / 10240 x 186000 miles = 64 / 10240 x 186000 x 5280 feet = 6138x10^3 feet

Therefore, 64 bytes is equal to 6138 x 10^3 feet long message

the conscious process of planning and controlling how you spend your time is called?

Answers

Hello There!

The conscious process of planning and controlling how you spend your time is called time management.
Time management is extremely important to get everything done. Being organized is a good skill.

Hope This Helps You!
Good Luck :) 

- Hannah ❤

Time Management is the conscious process of planning and controlling how you spend your time.

It involves organizing and planning your tasks and activities to make the most efficient use of your time. By prioritizing tasks, setting goals, and allocating time appropriately, you can manage your workload effectively and achieve a balance between work, leisure, and other activities. This process helps in enhancing productivity, reducing stress, and ensuring that important tasks are completed within set deadlines.

Time management skills are crucial for both personal and professional success, enabling individuals to handle multiple responsibilities and commitments efficiently.

Suppose that a system uses 16-bit memory words and its memory is built from 32 1m à 8 ram chips. how many address bits are required to uniquely identify each memory word

Answers

3 Ram is required for it to be honest

A _______ is used to analyze and summarize your data without graphical support.

Answers

Pivot table is the answer

Vaporization is the process in which liquid is sufficiently cooled to change states of matter from a liquid to a vapor true or false

Answers

Ans : The statement is False. Correct definition is Vaporization (or vapourisation) of an element or compound is a phase transition from the liquid phase to vapor. There are two types of vaporization: (1) evaporation and (2)boiling, not cooling. In the given question the statement is false. Evaporation is a surface phenomenon, whereas boiling is a bulk phenomenon.

Final answer:

The statement is false. Vaporization is the process in which a liquid is sufficiently heated, not cooled, to change states of matter from a liquid to a vapor.

Explanation:

Vaporization, also known as boiling or evaporation when below the boiling point, is the process where a liquid changes into a gas. This occurs when the molecules in a liquid gain enough energy to overcome the forces of attraction holding them together and enter the gaseous phase. Heat transfer from the surroundings to the liquid is necessary for vaporization to occur. On the other hand, condensation is the reverse process where a gas turns back into a liquid, which involves cooling and the release of heat to the surroundings. For example, when water boils, it reaches its vaporization temperature, allowing it to convert to a gas, with the amount of heat required described by the equation Q = mLv, where Lv is the latent heat of vaporization.

When working in Excel Online, what Ribbon has the command to add a hyperlink?

Home

Insert

Review

View

Answers

It should be Insert unless your settings aren't default...

The correct answer is B) the insert tab

The option of _____ displays a dialog box that provides all the different paste options.

Answers

In which application? When you right click usually you are provided the context menu in a large number of applications which provides cut, copy, & paste options amongst others.

Why was the ipv6 protocol created for ip addresses?

Answers

The most obvious improvement in IPv6 over IPv4 is that IP addresses are lengthened from 32 bits to 128 bits. This extension anticipates considerable future growth of the Internet and provides relief for what was perceived as an impending shortage of network addresses. IPv6 also supports auto-configuration to help correct most of the shortcomings in version 4, and it has integrated security and mobility features.

The original amount of money you deposit in a savings account is called __

Answers

it is called principal 

Consider ________ when designing for display on a mobile device. font size all of these contrast small screen size

Answers

to be honest...i really dont know 

Answer:

all of these

Explanation:

    Good application design is one that considers the knowledge the consumer already has. In this regard, application designs that value the platforms for which they were developed are valued and use conventions that are already rooted in people's minds. That's why it's important to consider color patterns, typography, responsive design, and platforms when developing an application.

d. Convert binary fraction .10112 to Decimal fraction.

Answers

The binary fraction .10112 is converted to the decimal fraction 0.687510 by multiplying each binary digit by the corresponding power of 2 and summing the results.

To convert the binary fraction .10112 to a decimal fraction, we need to multiply each binary digit by the corresponding power of 2 based on its position after the decimal point. The first digit after the binary point stands for 2-1, the second for 2-2, the third for 2-3, and so forth.

Here's how this conversion is done:

The first '1' in .10112 is [tex]1 x 2^-1[/tex] = 0.5

The next '0' is 0 x 2-2 = 0

The following '1' is [tex]1 x 2^-3[/tex] = 0.125

The last '1' is [tex]1 x 2^-4[/tex] = 0.0625

Add these values together to get the decimal equivalent:

0.5 + 0 + 0.125 + 0.0625 = 0.687510

So, the binary fraction .10112 equals the decimal fraction 0.6875.

Semiconductor memory is used mainly for primary storage even with its high cost. In another hand, the magnetic tape is the cheapest storage media and it is commonly preferred only as backup storage! Use your own words to discuss how the storage cost, speed and capacity trade-offs can affect the choice of storage media

Answers

In general you want to use the cheapest storage medium who's speed is compatible with the function it needs to perform. For active storage that's handled while running programs, you need memory whose speed is closely matched to the processor speed. And that would be the rather expensive semiconductor memory which is close to ideal for the task. But semiconductor memory has the disadvantage of being expensive and it loses the values stored when power is lost. So slower, persistent storage is used such as SSD (Solid State Drives) and hard disks. That media is cheaper, but slower, but still fast enough to handle tasks such as loading programs and data into memory for execution, or storing data generated by programs to persistent storage. But as with all man made things, disasters happen. Computers break down, hard disks crash, floods and fires happen, etc., and as a result data is lost. So we make backups. Backups have to have a lot of storage and they have to be cheap. But they don't need rapid access, you can start at the beginning and read (or write) all the way to the end. And for that purpose, magnetic tape is ideal. Magnetic tape is actually quite fast when you're simply streaming a continuous stream of data without any need to randomly access any piece of that data. And it's cheap, so you're willing to make a back up copy of your system and store that backup off site so a single disaster won't destroy both the primary system and the backup.
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