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 (B)</p> Signup and view all the answers

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

<p>Sunday (D)</p> Signup and view all the answers

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

<p>Monday (B)</p> Signup and view all the answers

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

<p>Sunday (C)</p> Signup and view all the answers

How many leap years are between 2013 and 2022?

<p>2 (D)</p> Signup and view all the answers

What is the remainder when 150 is divided by 12?

<p>2 (B)</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 (B)</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 (D)</p> Signup and view all the answers

What is the remainder when 211 is divided by 7?

<p>1 (B)</p> Signup and view all the answers

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

<p>Wednesday (D)</p> Signup and view all the answers

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

<p>10 AM (D)</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 (D)</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 (A)</p> Signup and view all the answers

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

<p>Thursday (D)</p> Signup and view all the answers

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

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

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

<p>4 leap years (C)</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 (C)</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 (A)</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 (C)</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 (B)</p> Signup and view all the answers

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

<p>2030 (D)</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 (A)</p> Signup and view all the answers

What is the quotient when dividing 100 by 12?

<p>8 (C)</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 (B)</p> Signup and view all the answers

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

<p>11:00 PM (C)</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 (D)</p> Signup and view all the answers

What is the remainder when dividing 14 by 12?

<p>2 (C)</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 (C)</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 (C)</p> Signup and view all the answers

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

<p>True (C)</p> Signup and view all the answers

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

<p>True (B)</p> Signup and view all the answers

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

<p>False (D)</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 (C)</p> Signup and view all the answers

Which of the following is NOT an example of congruence?

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

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

<p>51 (A)</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 (B)</p> Signup and view all the answers

What can be inferred about the modulo operation?

<p>It checks the remainder of division by n (D)</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. (C)</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. (A)</p> Signup and view all the answers

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

<p>25 and 3 (A)</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. (D)</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. (D)</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 $2$. (A)</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. (D)</p> Signup and view all the answers

Which of the following examples illustrates congruence modulo 3?

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

Flashcards

Finding a Day of the Week

To determine the day of the week for a date, we divide the number of days by 7 and get the remainder. Here, the remainder represents the number of days we must move forward from the reference day.

Modulo Operation

The remainder after a division operation, denoting how many times a number fits into another number, is called the modulo.

Modular Arithmetic

A system where numbers 'wrap around' or reset after a certain limit. In this system, the focus is on remainders rather than the actual result of division.

Keyword 'Ago'

The keyword 'ago' indicates we need to move backward from the given day.

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Moving Backward

If the keyword is 'ago', we move backward from the reference day. If the remainder is 2 after dividing by 7, we move 2 days backward.

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Modular Arithmetic for Time

The modulo operation applied to time calculations. It uses a 12-hour clock system, meaning hours 'wrap around' every 12 hours.

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Keyword 'After'

The keyword 'after' indicates we need to move forward from the given day.

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Calculating Time Before/After

The process of finding the time a certain number of hours before or after a given time. This involves calculating the modulo of the number of hours and the 12-hour clock cycle.

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Modulo for Time Calculation

The result of the modulo operation for the time calculation, representing the remaining hours after complete 12-hour cycles, determines the time on a 12-hour clock.

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Moving Forward

If the keyword is 'after', we move forward from the reference day. If the remainder is 0 after dividing by 7, we remain on the same day.

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Automatic Date Move

In a non-leap year, a day on a specific date will automatically move one day forward in the next year.

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Quotient for Time Shift

The direction of the time shift, whether it's before or after the given time, depends on the quotient obtained from dividing the number of hours by 12.

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Leap Year Extra Move

Leap years require an extra day move because they have an extra day in February.

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Total Day Movements

To calculate the total number of day movements, we add the automatic moves and extra moves due to leap years.

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Quotient (Time Calculation)

The result of dividing the number of hours by 12.

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Shift (Time Calculation)

Indicates whether the time reference for the answer remains the same (e.g., AM or PM) as the given time or changes (e.g., AM to PM).

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Remainder (Time Calculation)

The remaining number of hours after dividing by 12.

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Even Quotient Rule (Time Calculation)

If the quotient (result of division) is even, the final reference time for the answer will be the same as the given time (e.g., both AM or both PM).

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Odd Quotient Rule (Time Calculation)

If the quotient (result of division) is odd, the final reference time for the answer will change from the given time (e.g., AM to PM or PM to AM).

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Adjusting the Reference Time (Time Calculation)

The process of adding or subtracting the remainder from the reference time to find the final answer.

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Day Calculation (16 Days)

The day of the week that is 16 days after Monday.

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Final Day (Day Calculation)

The day of the week, after taking into account full weeks (multiples of 7 days), determined by the remainder after division.

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Leap Year

A year in which an extra day is added to the month of February to compensate for the discrepancy between Earth's revolution around the sun and the calendar year.

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Day of the Week

The day of the week that a specific date falls on in a given year. For example, February 14, 2021, falls on a Sunday.

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Automatic Move

The number of days in a year, excluding leap years. It adds one day to the day of the week for each year.

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Extra Move

The number of days added to the day of the week due to leap years occurring within the given time interval. These days are added in addition to the 'Automatic Move'.

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Total Move

A calculated number of days moved forward from a starting date to arrive at the day of the week for a specific date in a future year. It includes both 'Automatic Moves' and 'Extra Moves'.

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Finding the day of the week in future year

Determining the day of the week for a specific date in a future year by sequentially adding 'Automatic Moves' and 'Extra Moves' to the original day of the week and dividing by 7. The remainder then indicates the number of days to shift forward from the starting day.

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Year Interval

The process of calculating the total number of years between two years, used for determining days of the week.

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Counting Leap Years

Counting the total number of leap years that fall within the specified year interval. This number is used for calculating the 'Extra Move'.

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Modulo Congruence Definition

Two integers, 'a' and 'b', are said to be congruent modulo 'n', where 'n' is a natural number, if the difference between 'a' and 'b' divided by 'n' results in an integer.

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Congruence Symbol (≡)

The symbol used to express that two integers 'a' and 'b' are congruent modulo 'n'.

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Modulus (n)

The natural number 'n' that acts as the divisor in modulo congruence. It defines the 'cycle' or 'modulus' of the congruence.

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Congruence (a ≡ b mod n)

The result of applying modulo congruence, where the difference between 'a' and 'b' is divisible by 'n'.

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True Congruence Statement

A statement in modulo congruence that is true if the given integers are congruent modulo 'n'.

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False Congruence Statement

A statement in modulo congruence that is false if the given integers are not congruent modulo 'n'.

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Result of Modulo Congruence

An integer resulting from the difference between the two integers (a and b) divided by the modulus 'n'.

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Determining Congruence

The result of modulo congruence is an integer, indicating the two numbers are congruent. If the result contains a fractional part, it indicates a remainder and the numbers are not congruent.

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Congruence modulo n

Two integers are congruent modulo n if their difference is divisible by n.

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Congruence symbol

The symbol ≡ indicates that two integers are congruent modulo a given number.

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Check for congruence

If a-b is divisible by n, then a is congruent to b modulo n.

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Applications of Modulo

The modulo operation is frequently used in computer science, especially in fields like cryptography and hashing.

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Congruence notation

The expression 'a≡b mod n' represents the statement that 'a is congruent to b modulo n'.

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Congruence check

Checking if a number is congruent to another modulo a specific number can be done by dividing the difference between the two numbers by the modulo.

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Modulo and patterns

The modulo operation is useful for understanding the behavior of repeating patterns and cycles in various mathematical contexts.

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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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