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Chapter 4: Problem 16

What time does a 24 -hour clock read a) 100 hours after it reads \(2 : 00 ?\) b) 45 hours before it reads \(12 : 00 ?\) c) 168 hours after it reads \(19 : 00 ?\)

Short Answer

Expert verified
a) 6:00, b) 15:00, c) 19:00

Step by step solution

01

- Understanding the 24-hour clock

A 24-hour clock resets after every 24 hours. This means that 24:00 is the same as 00:00.
02

- Calculate 100 hours after 2:00

Calculate the remainder when 100 is divided by 24 to find the effective hours: 100 mod 24 = 4. Add these 4 hours to 2:00. Therefore, 2:00 + 4 hours = 6:00.
03

- Calculate 45 hours before 12:00

Calculate the remainder when 45 is divided by 24 to find the effective hours: 45 mod 24 = 21. Subtract these 21 hours from 12:00. Therefore, 12:00 - 21 hours = 15:00 on the previous day.
04

- Calculate 168 hours after 19:00

Calculate the remainder when 168 is divided by 24 to find the effective hours: 168 mod 24 = 0. Since the remainder is 0, 168 hours after 19:00 is exactly the same time, 19:00.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

modular arithmetic
Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, called the modulus.
In the context of a 24-hour clock, the modulus is 24. This is because the clock resets after every 24 hours, making 24:00 the same as 00:00.
For example, in modular arithmetic, if you add 26 hours to a given time, you can calculate the equivalent time by finding the remainder of 26 divided by 24. Here, 26 mod 24 gives a remainder of 2. Thus, 26 hours later is the same as 2 hours later on a 24-hour clock.
This property of modular arithmetic makes it ideal for dealing with continuous cycles like hours in a day.
time calculation
Time calculation using a 24-hour clock can sometimes seem confusing, but modular arithmetic makes it easier.
Anytime you need to figure out a time after or before a certain number of hours, you can use the modulus operation to simplify the process.
For example, in the given problem:
  • To find the time 100 hours after it reads 2:00, you calculate 100 hours mod 24. Here, 100 mod 24 equals 4, meaning 100 hours is effectively 4 hours later. Adding these 4 hours to 2:00, you get 6:00.
  • To find the time 45 hours before it reads 12:00, you calculate 45 hours mod 24. Here, 45 mod 24 equals 21, meaning 45 hours before is effectively 21 hours earlier. So, subtracting 21 hours from 12:00 gives you 15:00 the previous day.
  • For 168 hours after 19:00, 168 hours mod 24 equals 0 since 168 is exactly a multiple of 24. Thus, 168 hours later is the same as 0 hours later, meaning it is still 19:00.
These simplified time calculations help avoid the confusion of counting hours manually.
problem-solving steps
Follow these steps to solve 24-hour clock problems easily:
1. **Understand the 24-hour clock:** Realize it resets after every 24 hours.
2. **Use modular arithmetic:** This helps to find the effective hours by calculating the remainder when dividing by 24.
3. **Addition and subtraction:** Add the effective hours if you're moving forward in time, or subtract them if moving backward.
4. **Verify your solution:** Ensure the result falls within the 24-hour cycle.
By systematically applying these steps, you can solve any problem related to hours on a 24-hour clock effectively.
remainders
Remainders play a crucial role in modular arithmetic and, consequently, in solving 24-hour clock problems. The remainder is what鈥檚 left over after division.
For instance:
  • If you want to find 100 hours after 2:00, divide 100 by 24; the remainder is 4. This remainder tells you the effective hours you need to move forward.
  • Similarly, to find 45 hours before 12:00, divide 45 by 24; the remainder is 21. This remainder tells you the effective hours to move backward.
  • Finally, for 168 hours after 19:00, since 168 is a multiple of 24, the remainder is 0. This tells you the time doesn鈥檛 change.
Remainders help you correctly adjust time calculations within the constraints of a 24-hour cycle. This is fundamental for accurate time computation in real-life applications.

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Most popular questions from this chapter

Decrypt these messages encrypted using the shift cipher \(f(p)=(p+10) \bmod 26 .\) a) CEBBOXNOB XYG b) LO WIPBSOXN c) DSWO PYB PEX

In Exercises \(31-32\) suppose that Alice and Bob have these public keys and corresponding private keys: \(\left(n_{\text { Alice }}, e_{\text { Alice }}\right)=\) \((2867,7)=(61 \cdot 47,7), \quad d_{\text { Alice }}=1183\) and \(\left(n_{\text { Bob }}, e_{\text { Bob }}\right)=\) \((3127,21)=(59 \cdot 53,21), d_{\text { Bob }}=1149 .\) First express your answers without carrying out the calculations. Then, using a computational aid, if available, perform the calculation to get the numerical answers. Alice wants to send to all her friends, including Bob, the message "SELL EVERYTHING" so that he knows that she sent it. What should she send to her friends, assuming she signs the message using the RSA cryptosystem.

Describe the steps that Alice and Bob follow when they use the Diffie-Hellman key exchange protocol to generate a shared key. Assume that they use the prime \(p=23\) and take \(a=5,\) which is a primitive root of \(23,\) and that Alice selects \(k_{1}=8\) and Bob selects \(k_{2}=5 .\) (You may want to use some computational aid.)

What are the greatest common divisors of these pairs of integers? a) \(3^{7} \cdot 5^{3} \cdot 7^{3}, 2^{11} \cdot 3^{5} \cdot 5^{9}\) b) \(11 \cdot 13 \cdot 17,2^{9} \cdot 3^{7} \cdot 5^{5} \cdot 7^{3}\) c) \(23^{31}, 23^{17}\) d) \(41 \cdot 43 \cdot 53,41 \cdot 43 \cdot 53\) e) \(3^{13} \cdot 5^{17}, 2^{12} \cdot 7^{21}\) f) \(1111,0\)

Show that 2 is a primitive root of 19
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