Modular Arithmetic and Time Calculations
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Questions and Answers

What will be the time 100 hours before 1:00 PM if the quotient is even?

  • 8 AM
  • 11 AM
  • 1 PM
  • 9 AM (correct)

    • After determining a reference time from the quotient, what does an odd quotient indicate for the shift?

  • The shift stays the same
  • The shift changes (correct)
  • The shift is unchanged
  • No shift occurs

    • If today is Saturday, what day was it 211 days ago?

  • Wednesday
  • Friday (correct)
  • Monday
  • Thursday

    • What is the reference time when calculating 247 hours after 3:00 PM?

    <p>3 PM</p> Signup and view all the answers

    What day will it be after 147 days if today is Sunday?

    <p>Sunday</p> Signup and view all the answers

    If May 19, 2013 was a Thursday, what day will it fall on in 2022?

    <p>Monday</p> Signup and view all the answers

    If today is Monday, what day will it be 16 days from today?

    <p>Sunday</p> Signup and view all the answers

    How many leap years are between 2013 and 2022?

    <p>2</p> Signup and view all the answers

    What is the remainder when 150 is divided by 12?

    <p>2</p> Signup and view all the answers

    For a quotient of 10 after dividing by 12, what can be inferred about the shift?

    <p>The shift will remain the same</p> Signup and view all the answers

    If your birthday falls on a Monday this year, what day will it fall on next year?

    <p>Tuesday</p> Signup and view all the answers

    What is the remainder when 211 is divided by 7?

    <p>1</p> Signup and view all the answers

    If today is Tuesday, which day will it be 20 days from today?

    <p>Wednesday</p> Signup and view all the answers

    What will happen to the time if moving backwards 8 hours from 5 PM?

    <p>10 AM</p> Signup and view all the answers

    If today is Wednesday and you need to find out the day 10 days from now, what day will it be?

    <p>Tuesday</p> Signup and view all the answers

    If today is Thursday and it is a leap year, what day will it fall on next year?

    <p>Friday</p> Signup and view all the answers

    If February 14, 2021 is a Sunday, what day will it fall on in 2036?

    <p>Thursday</p> Signup and view all the answers

    How many years are there in the interval from 2013 to 2022?

    <p>7 years</p> Signup and view all the answers

    How many leap years are in the interval from 2024 to 2036?

    <p>4 leap years</p> Signup and view all the answers

    If Independence Day is celebrated on the second Friday of June 2021, what day will it fall on in 2041?

    <p>Tuesday</p> Signup and view all the answers

    What is the result when the total of automatic moves and extra moves is divided by 7 in the context of finding the day for Independence Day in 2041?

    <p>3 remainder 4</p> Signup and view all the answers

    How many extra moves are counted in the interval including the leap years up to 2040?

    <p>5 extra moves</p> Signup and view all the answers

    How do you determine the day of the week based on the calculated moves?

    <p>Add the remainder to the starting day of the week</p> Signup and view all the answers

    Which year is not a leap year within the interval of 2024 to 2040?

    <p>2030</p> Signup and view all the answers

    How many hours after 4:00 AM is it if 14 hours are added?

    <p>6:00 PM</p> Signup and view all the answers

    What is the quotient when dividing 100 by 12?

    <p>8</p> Signup and view all the answers

    If it is currently 10:00 PM and you set an alarm for 7 hours later, what time will it be when the alarm goes off?

    <p>4:00 AM</p> Signup and view all the answers

    What will be the time 100 hours before 1:00 PM?

    <p>11:00 PM</p> Signup and view all the answers

    When adding hours to a time, what must be considered to determine if the time shifts from AM to PM?

    <p>The quotient of the division by 12</p> Signup and view all the answers

    What is the remainder when dividing 14 by 12?

    <p>2</p> Signup and view all the answers

    If the clock shows 4:00 PM after adding 12 hours to 4:00 AM, what time would it represent?

    <p>4:00 PM</p> Signup and view all the answers

    What is the time after 7 hours if it starts at 10:00 PM?

    <p>5:00 AM</p> Signup and view all the answers

    Is the statement $48 \equiv 3 \mod 5$ true or false?

    <p>True</p> Signup and view all the answers

    Determine the validity of the statement $76 \equiv 6 \mod 7$.

    <p>True</p> Signup and view all the answers

    Is $59 \equiv 8 \mod 9$ a true statement?

    <p>False</p> Signup and view all the answers

    Which of the following defines two integers a and b as congruent modulo n?

    <p>If $a = b + kn$ for some integer k</p> Signup and view all the answers

    Which of the following is NOT an example of congruence?

    <p>$15 \equiv 10 \mod 6$</p> Signup and view all the answers

    What is the result of $59 - 8$ as it relates to modulo 9?

    <p>51</p> Signup and view all the answers

    In the context of the equation $a \equiv b \mod n$, what does 'n' represent?

    <p>A natural number</p> Signup and view all the answers

    What can be inferred about the modulo operation?

    <p>It checks the remainder of division by n</p> Signup and view all the answers

    Which of the following statements is true regarding the congruence relation?

    <p>If $a ≡ b \mod n$, then $a - b$ divided by $n$ must result in an integer.</p> Signup and view all the answers

    What is the correct statement about the congruence $11 ≡ 11 \mod 2$?

    <p>It is true because $11 - 11 = 0$ is an integer.</p> Signup and view all the answers

    Which of the following pairs of numbers is NOT congruent modulo 7?

    <p>25 and 3</p> Signup and view all the answers

    What criterion must be satisfied for two integers to be congruent modulo n?

    <p>The difference $a - b$ must result in a quotient that is an integer when divided by n.</p> Signup and view all the answers

    Which of the following is the result of the congruence $30 ≡ 5 \mod n$ for $n = 25$?

    <p>It is true since the difference $30 - 5 = 25$ is divisible by 25.</p> Signup and view all the answers

    Is the statement $7 ≡ 5 \mod 2$ true or false?

    <p>True, because $7 - 5 = 2$ and $2$ is divisible by $2$.</p> Signup and view all the answers

    If $a ≡ b \mod n$, which of the following can be inferred?

    <p>a and b have the same remainder when divided by n.</p> Signup and view all the answers

    Which of the following examples illustrates congruence modulo 3?

    <p>4 ≡ 1 \mod 3</p> Signup and view all the answers

    Study Notes

    Modular Arithmetic and Applications

    • Modular arithmetic focuses on the remainder when dividing one number by another
    • The modulo operation finds the remainder
    • Finding the time after a specified number of hours involves dividing the hours by 12, and using the remainder to determine the time.

    Time Calculations

    • To find the time 14 hours after 4:00 AM, divide 14 by 12. The remainder is 2. Since 12 hours after 4 AM is 4 PM, add 2 hours to get 6 PM.

    • To calculate 100 hours before 1:00 PM, divide 100 by 12. The remainder is 4. The quotient (8) is even, meaning the reference point (PM) is unchanged. Moving 4 hours back from 1 PM results in 9 AM.

    • To find the time 247 hours after 3:00 PM, divide 247 by 12. The remainder is 7. Since the quotient (20) is even, the reference time (PM) remains the same. Add 7 hours to 3 PM to get 10 PM.

    Day Calculations

    • To determine the day of the week after a given number of days, divide the number of days by 7. The remainder indicates the number of days to add/subtract from the reference day.

    • If today is Monday, what day will it be 16 days from now? 16÷7 results in a remainder of 2. Add 2 days to Monday to get Wednesday.

    • If today is Saturday, what was the day 211 days ago? 211÷7 results in a remainder of 1. Subtract 1 day from Saturday to get Friday.

    • If today is Sunday, what day will it be 147 days later? 147÷7 = 21 with a remainder of 0. The day will remain Sunday.

    Date Calculations

    • Days of the week shift forward one day each year, except during leap years.

    • A leap year has an extra day in February (29 days), which affects the day of the week calculations. To adjust for leap years, count the leap years in the given number of years

    • If May 19, 2013, was a Thursday, what day will it be in 2022? There are 9 years between 2013 and 2022. Counting the leap years in the interval results in adding an extra two days. Thus, the total shift is 11 days. Adding 11 days to Thursday, results in Monday.

    Modulus Congruence

    • Two integers 'a' and 'b' are congruent modulo 'n' (written as a ≡ b (mod n)) if the difference (a − b) is divisible by 'n' (i.e., (a − b)/n is an integer)

    • The modulus 'n' is a positive integer.

    • Examples:

      • 11 ≡ 1 (mod 2) because (11 − 1)/2 = 5
      • 48 ≡ 3 (mod 5) because (48 − 3)/5 = 9
      • 76 ≡ 6 (mod 7) because (76 − 6)/7 = 10

    • 25 ≢ 3 (mod 7) because (25 − 3)/7 is not an integer

    • 59 ≢ 8 (mod 9) because (59 − 8)/9 is not an integer

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    Related Documents

    Modular Arithmetic PDF

    Description

    This quiz explores the principles of modular arithmetic and its applications in calculating time and day. Participants will learn how to efficiently determine time and day changes based on simple calculations. Understanding the modulo operation is essential for mastering these concepts.

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