Modular Arithmetic

Lecture 11

Modular arithmetic is arithmetic in which numbers do

not continue forever. Modulo 7 has numbers 0, 1, 2, 3, 4, 5, and 6. Modulo 5 has numbers 0, 1, 2, 3, and 4. In general, modulo n has numbers 0, 1, 2, …, n-1.

Modular Basics

Clocks start over at 12 and/or 24 hours. Calendars start over after

7 days for one week 52 weeks for one year 365 days for one year

You must pay attention to going forward in time and going backward in time in order to add or subtract from where you are.

Clocks and Calendars

Class starts at 2:30 PM on Thursday.

If I assign a paper due 145 hours from the start of class, when will it be due?

If you had a paper due that was assigned 80 hours prior, when was it assigned?

Clock Examples

If we divide 145 hours by 24 (hours in a day), we find 6 whole days and a decimal.

Subtracting 6 from our quotient we find 0.041666667 remaining. That is, 0.0416666 of a 24 hour period.

Multiply this decimal portion by 24 to see how many hours it represents – 24(0.0416666667) = 1.

That is, this assignment is due 145 hours, or 6 days and 1 hour from the start of class. It would be Wednesday at 3:30 PM.

145 hours after 2:30 PM

Once again we divide 80 by 24 to get 3.333333.

This means three days prior and 0.333333 of 24 hours…24(0.3333333) = 8 hours.

Three days prior to Thursday is Monday and 8 hours prior to 2:30 is 6:30 AM.

This assignment was given at 6:30 AM on Monday and due 80 hours later on Thursday at 2:30 PM.

80 hours prior to 2:30 PM

April 20th, 2014 falls on a Sunday.

What day of the week was 4/20 in 1980? What is the next year that 4/20 falls on a Sunday?

Calendar Examples

The first thing we should do is determine which years

were leap years (366 days) and which were not (365 days).

Leap day occurs in any year that is evenly divisible by 4 (with an exception that doesn’t arise in our example).

Leap years between 1980 and now were in 1980, 1984, 1988, 1992, 1996, 2000, 2004, 2008, and 2012.

4/20/1980

We know that from April 20th to April 20th is a year.

We can count the whole years and the leap years, multiplying appropriately to find the number of days.

1980 is a leap year, but April 20th is after leap day so it won’t count. There are 8 leap years and 26 non-leap years.

This makes 366x8=2928 and 26x365=9490 days, respectively, for a total of 12, 418 days.

In order to determine the day of the week, we now

divide this number of days by 7 to get 1774 with no decimal.

No decimal tells us it was exactly 1774 weeks prior…4/20 fell on a Sunday in 1980.

In 2015, it will be a Monday. 2016 is a leap year so will be Wednesday. 2017 will be Thursday. 2018 will be Friday. 2019 will be Saturday. 2020 is a leap year so will be a Monday… D’oh! 2021 Tuesday, 2022 Wednesday, 2023 Thursday,

2024 Saturday, 2025 will be the next time 4/20 falls on a Sunday.

4/20 next time on Sunday

Check digits are a system for us to make sure the

input numbers are correct in a variety of sources. Check digits allow us to detect errors through a

variety of systems.

Check Digits