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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
4 years ago

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16 votes

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  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?

Mark Hennings - 4 years ago

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

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

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.

Michael Tong - 4 years ago

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

Calvin Lin Staff - 4 years ago

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@Calvin Lin 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.

Mark Hennings - 4 years ago

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@Mark Hennings 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.

Calvin Lin Staff - 4 years ago

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What are the last three digits of \( 9^{9^9} \)?

Michael Tong - 4 years ago

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

Mark Hennings - 4 years ago

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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?

Michael Tong - 4 years ago

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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 \(6\).

Parth Chopra - 4 years ago

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

Bob Krueger - 4 years ago

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

Calvin Lin Staff - 4 years ago

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

Michael Tong - 4 years ago

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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.

Anton Than Trong - 4 years ago

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

Wahyu Adi Saputro - 4 years ago

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

Wahyu Adi Saputro - 4 years ago

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What are the last three digits of \(2003^{2002^{2001}}\)?

Tong Zou - 3 years, 11 months ago

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

Philips Zephirum Lam - 4 years ago

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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?

Daniel Liu - 4 years ago

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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.

Calvin Lin Staff - 4 years ago

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

Bob Krueger - 4 years ago

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

Michael Tong - 4 years ago

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@Bob Krueger 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.

Calvin Lin Staff - 4 years ago

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