Modular Arithmetic

This week, we learn about Modular Arithmetic, a system of arithmetic for the integers with many applications.

How would you use Modular Arithmetic to solve the following?

1. In the years 1600-1999, how many times was the first of January celebrated on a Sunday?

2. Share a question which can easily be approached by Modular Arithmetic.

Note by Calvin Lin
6 years, 7 months ago

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What are the last three digits of $9^{9^9}$?

- 6 years, 7 months ago

We have $9^{50} \equiv 1$ ($1000$) and $9^9 \equiv 39$ ($50$). Hence $9^{9^9} \equiv 9^{39} \equiv 289$ ($1000$).

- 6 years, 7 months ago

1. Given any year, find the date of Easter (assuming the Gregorian Calendar!).

2. Is there a positive integer $n$ for which $n^7 - 77$ is a Fibonacci number?

- 6 years, 7 months ago

$n^7 \equiv 0, 1, 12, 17, 28 \pmod {29}$

$n^7 - 77 \equiv 9, 10, 11, 22, 27 \pmod {29}$

The pisano period of $29$ is $1, 1, 2, 3, 5, 8, 13, 21, 5, 26, 2, 28, 1, 0$, none of the values are the same.

- 6 years, 7 months ago

Can you explain why you chose modulo 29? That seems like such a random number.

Staff - 6 years, 7 months ago

Seven $7$th roots of unity modulo $29$, so only $1+4=5$ seventh powers, which gives a fighting chance of the residues for $n^7-77$ being distinct from the Fibonacci residues.

- 6 years, 7 months ago

Indeed. A clearer explanation of the motivation, is to say that if $p$ is a prime of the form $7k + 1$, then there are $k + 1$ seventh powers (where the +1 accounts for 0 ). This gives a fighting change of the residues being distinct from the Fibonacci residues. So, we try the smallest prime of the form $7k+1$, which is 29. Had it not worked, we will look at 43, so on and so forth.

Writing it this way takes away the mysticism of "why 29", and provides a path to approach the more general problem.

Staff - 6 years, 7 months ago

Here are a few easy ones I know about modular arithmetic:

The fractional part of $\frac {n}{7}$ is $.142857$ repeating, or some variant thereof. What is the minimal positive integer N such that the fractional part of $\frac{251}{7} \times n$ is $.142857$ repeating?

- 6 years, 7 months ago

Nice questions. I took the liberty of removing the answers and splitting out the questions, so that people can comment on them individually.

Staff - 6 years, 7 months ago

Sorry, I'm a new member, so I don't really know the formatting that well yet. We can start this problem by thinking about the number $251$, which can be written as $251 \equiv 6 \pmod{7}$. In order for $\frac {251}{7} \times n$ to be $0.142857$ repeating, $251n \equiv 1 \pmod{7}$. Looking at the first congruence, we can start by realizing that $251n$ will have a remainder of $1$ iff $6n \equiv 1 \pmod{7}$. We can see that the possible values for $n$ will be of the form $7k - 1$, where $k$ is a positive integer. Therefore, the minimum possible value for $n$ is $6$.

- 6 years, 7 months ago

Your formatting is fine. Note, though, that you could simplify things a bit by making $251 \equiv -1 \pmod 7$.

- 6 years, 7 months ago

Find all perfect squares such that they are divisible by 2 and not 4.

- 6 years, 7 months ago

Nothing, because the square root of 4 is 2 and 2 square is 4.

- 6 years, 7 months ago

nothing, it is impossible. Because, square root of 4 is 2, and 2 square is 4.

- 6 years, 7 months ago

All even numbers squared give an even number squared. An even number can be expressed by 2n, where n is any integer. Thus, an even number squared can be expressed as 2^2n^2, which is 4n^2. Thus, no perfect squares are divisible by 2 and not 4.

- 6 years, 7 months ago

For the first question, are we assuming that leap years happen every 4 years, or are we following the leap year rules and omitting the leap year on 1700, 1800, and 1900?

- 6 years, 7 months ago

Ah, you got me. I was hoping that would fly beneath the radar.

To answer this question well, you have to accurately account for all of the leap years, and their effects on the calendar.

Staff - 6 years, 7 months ago

But an even more trick question: are we supposed to accommodate for the 1752 calendar change?

- 6 years, 7 months ago

The Gregorian calendar was implemented in 1582 (which was why I chose 1600).

Not my fault that Britain (and her colonies) were late to the party.

FYI: The gregorian calendar was adopted at different points in time. Note that not everyone use the gregorian calendar.

Staff - 6 years, 7 months ago

No, the calendar change is trivial, and is left as an exercise to the reader.

- 6 years, 7 months ago

Reply to Calvin, 1. In a 400-year cycle, there are 58 years was the first of January celebrated on a Sunday.

- 6 years, 7 months ago

What are the last three digits of $2003^{2002^{2001}}$?

- 6 years, 5 months ago