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Modular Arithmetic (Congruences) Of Elementary Number Theory : Lucid Explanation With Examples

The eminent mathematician Gauss, who is considered as one of the greatest in history has quoted "mathematics is the queen of sciences and number theory is the queen of mathematics."

Several important discoveries of Elementary Number Theory such as Fermat's little theorem, Euler's theorem, the Chinese remainder theorem are based on simple arithmetic of remainders.

This arithmetic of remainders is called Modular Arithmetic or Congruences.

In this article, I aim to explain "Modular Arithmetic (Congruences)" in such a simple way, that a common man with little math back ground can also understand it.

I supplement the lucid explanation with examples from everyday life.

For students, who study Elementary Number Theory, in their under graduate or graduate courses, this article will serve as a simple introduction.

Modular Arithmetic (Congruences) of Elementary Number Theory :

We know, from the knowledge of Division

Dividend = Remainder + Quotient x Divisor.

If we denote dividend by a, Remainder by b, Quotient by k and Divisor by m, we get

a = b + km

or a = b + some multiple of m

or a and b differ by some multiples of m

or if you take away some multiples of m from a, it becomes b.

Taking away some (it does n't matter, how many) multiples of a number from another number to get a new number has some practical significance.

Example 1 :

For example, look at the question

Today is Sunday. What day will it be 200 days from now ?

How do we solve the above problem ?

We take away multiples of 7 from 200. We are interested in what remains after taking away the mutiples of 7.

We know 200 ÷ 7 gives quotient of 28 and remainder of 4 (since 200 = 28 x 7 + 4)

We are not interested in how many multiples are taken away.

i. e., We are not interested in the quotient.

We only want the remainder.

We get 4 when some (28) multiples of 7 are taken away from 200.

So, The question, "What day will it be 200 days from now ?"

now, becomes, "What day will it be 4 days from now ?"

Because, today is sunday, 4 days from now will be Thursday. Ans.

The point is, when, we are interested in taking away multiples of 7,

200 and 4 are the same for us.

Mathematically, we write this as

200 ? 4 (mod 7)

and read as 200 is congruent to 4 modulo 7.

The equation 200 ? 4 (mod 7) is called Congruence.

Here 7 is called Modulus and the process is called Modular Arithmetic.

Let us see one more example.

Example 2 :

It is 7 O' clock in the morning.

What time will it be 80 hours from now ?

We have to take away multiples of 24 from 80.

80 ÷ 24 gives a remainder of 8.

or 80 ? 8 (mod 24).

So, The time 80 hours from now is the same as the time 8 hours from now.

7 O' clock in the morning + 8 hours = 15 O' clock

= 3 O' clock in the evening [ since 15 ? 3 (mod 12) ].

Let us see one last example before we formally define Congruence.

Example 3 :

A person is facing East. He rotates 1260 degree anti-clockwise. In what direction, he is facing ?

We know, rotation of 360 degrees will bring him to the same position.

So, we have to remove multiples of 360 from 1260.

The remainder, when 1260 is divided by 360, is 180.

i. e., 1260 ? 180 (mod 360).

So, rotating 1260 degrees is same as rotating 180 degrees.

So, when he rotates 180 degrees anti-clockwise from east, he will face west direction. Ans.

Definition of Congruence :

Let a, b and m be any integers with m not zero, then we say a is congruent to b modulo m, if m divides (a - b) exactly without remainder.

We write this as a ? b (mod m).

Other ways of defining Congruence include :

(i) a is congruent to b modulo m, if a leaves a remainder of b when divided by m.

(ii) a is congruent to b modulo m, if a and b leave the same remainder when divided by m.

(iii) a is congruent to b modulo m, if a = b + km for some integer k.

In the three examples above, we have

200 ? 4 (mod 7); in example 1.

80 ? 8 (mod 24); 15 ? 3 (mod 12); in example 2.

1260 ? 180 (mod 360); in example 3.

We started our discussion with the process of division.

In division, we dealt with whole numbers only and also, the remainder, is always less than the divisor.

In Modular Arithmetic, we deal with integers (i.e. whole numbers + negative integers).

Also, when we write a ? b (mod m), b need not necessarily be less than a.

The three most important properties of congruences modulo m are:

The reflexive property :

If a is any integer, a ? a (mod m).

The symmetric property :

If a ? b (mod m), then b ? a (mod m).

The transitive property :

If a ? b (mod m) and b ? c (mod m), then a ? c (mod m).

Other properties :

If a, b, c and d, m, n are any integers with a ? b (mod m) and c ? d (mod m), then

a + c ? b + d (mod m)

a - c ? b - d (mod m)

ac ? bd (mod m)

(a)n ? bn (mod m)

If gcd(c,m) = 1 and ac ? bc (mod m), then a ? b (mod m)

Let us see one more (last) example, in which we apply the properties of congruences.

Example 4 :

Find the last decimal digit of 13^100.

Finding the last decimal digit of 13^100 is same as

finding the remainder when 13^100 is divided by 10.

We know 13 ? 3 (mod 10)

So, 13^100 ? 3^100 (mod 10) .....(i)

We know 3^2 ? -1 (mod 10)

So, (3^2)^50 ? (-1)^50 (mod 10)

So, 3^100 ? 1 (mod 10) .....(ii)

From (i) and (ii), we can say

last decimal digit of 13^100 is 1. Ans.

For more of Elementary Number Theory, go to

http://www.math-help-ace.com/Elementary-Number-Theory.html

About the Author

Name : kvln
Qualifications : B.Tech., M.S. (from IIT, Madras)
Has 14 years of teaching experience.
Love for math and love for teaching
makes me feel more than happy to help.
Has own math web site :

http://www.math-help-ace.com/

Due West School

Don't Make This Common Law School Mistake

Before I went to law school, I read a few of the common law school guides that most law students read. These guides promised me academic success as long as I followed the particular strategies that they described. I followed their advice for about half of my time at law school, and it's a good thing I didn't keep following it longer.

As I got wiser to the ways of law school, I realized that much of the advice given to law students is counterproductive. As I moved away from the advice found in mainstream law school guides, my grades, and more importantly my overall law school experience, improved markedly.

Towards the end of my the time at law school and even for some time after I graduated, my friends and I would discuss all the mistakes we had made by listening to the silly mainstream law school advice. We would often wonder out loud how this advice came to be, illogical and ineffective as much of it is.

Now that I know better after making many mistakes as a law student, I will share with you one common law school strategy that you should fervently avoid if you want to do your best in law school. Unsurprisingly, most law school guides encourage this exact strategy, to your academic, financial, and social detriment.

The strategy I am talking about is the use of hornbooks and commercial study aids to prepare for classes and exams. As a law student, you should do your best to avoid this common mistake, and here are a number of reasons why. First, studying from commercially prepared study aids is not an effective study strategy, because the material in these study aids differs from your course material. The way study aids present material will also differ from the way in which your professor wants you to approach and understand the course material. Contrary to popular belief, you can get all the information you need to do extremely well on exams from the casebook, relevant statutes, and class, without ever resorting to study aids. Furthermore, buying hornbooks multiplies the number of sources you feel that you must study, and will only make the learning process more confusing, frustrating, and ineffective.

Second, hornbooks and commercial study aids are expensive, and the cost of these aids adds up fast, especially if you believe that you need them for each of your classes. Being that these extraneous materials contain little, if anything, of value to your studies, you will do well to save your money for more worthwhile pursuits, such as vacations, electronics, or whatever it is that you like. And yes, the money you save on hornbooks can actually pay for a vacation, even a nice one.

Third, given that study aids have nothing to offer over the actual course materials in terms of valuable study material, they waste your time. Your time in law school is valuable, and you should try to be as efficient as possible by using that time to study the casebook and to relax. The less time you spend on useless study aids, the more time you will have for effective study and for relaxation, which will help you stay balanced and focused in law school, while avoiding burnout.

Finally, you should be aware that most professors are against the use of study aids (except those who write these aids), and they know what they're talking about. Your professors were exceptional law students, and most of the time they know what they're talking about when it comes to study strategy, so listen when they tell you that the most important information is in the casebook and statutory supplement, rather than in a commercial study aid.

Instead of wasting your time and money on hornbooks and commercial study aids, read your class materials and devote at least some of your free time to activities that you enjoy.

 

 

 

 

 

About the Author

Michael West is the author of Beat Law School: Unconventional Advice for Current and Future Law Students, available at http://www.amazon.com/gp/product/1452849315/

Due West Elementary School

West Ottawa – the City’s Trendiest Place to Live

West Ottawa – it’s perhaps the city’s trendiest area to live. But what exactly makes it the hot spot for Ottawa apartment seekers? Is it the large variety of shopping? Perhaps it’s the abundant green space, bike paths and breathtaking views of the Ottawa River? Or maybe it’s the fact that West Ottawa is brimming with culture, hosting arts and theatre festivals all year round? Regardless of the reason, it’s hardly a mystery why so many people choose to call a West End Ottawa apartment their home.

Paramount Properties has four apartment buildings for rent in Ottawa’s West End – Greenbank Towers, The Maples Apartments, Carling Park Apartments and Wellington Towers. Perfect for students, artists and outdoors enthusiasts, an apartment rental in West Ottawa has everything you need – and a lot more!

Transportation

The West End is easily accessible by bus via the Transitway – OC Transpo routes 95, 96 and 97 all pass through Lincoln Fields station (near Richmond and Carling). If you’re renting an Ottawa apartment near Lincoln Fields, you’re right beside one of the major transit hubs in the city, as the station connects dozens of local and citywide buses in one convenient location. If you have your own vehicle, both the Queensway and the Ottawa River Parkway are just minutes away, quickly connecting you to the rest of the city.

Major Shopping Centres

If you’re looking at an apartment for rent in Westboro or Nepean, you’ll be glad to know that those neighbourhoods are bustling with numerous shopping opportunities. In addition to the many small independent shops (including several outdoor sports stores), there are many larger shopping centres in the area, including Bayshore Shopping Centre. A three-level mall containing more than 150 stores and services, it is ideally located just off Highway 417. You should also check out Carlingwood Shopping Centre, which contains more than 125 stores and services with plenty of free parking.

Other West Ottawa malls include Lincoln Fields Shopping Centre (which contains a Wal-Mart and a Loeb), Pinecrest Shopping Centre (with the famous IKEA store overlooking the Queensway), College Square (a favourite destination for Algonquin College students and those with apartment rentals near Baseline and Woodroffe) and Westgate Shopping Centre (featuring a very useful 24-hour Shoppers Drug Mart).

Schools

A Paramount Properties West Ottawa apartment is perfect for students attending Algonquin College, as our Maples Apartments are within walking distance of the school and our Greenbank, Carling Park and Wellington apartments can quickly get you to campus by bus. And for those with young families, West Ottawa also has over 20 public elementary schools, 9 Catholic elementary schools, 5 public high schools and 2 Catholic high schools.

Museums and Art Centres

For those living in a Nepean apartment and looking to learn a little bit more about the history of their community, the Nepean Museum (16 Rowley Avenue), which is home to several antiques (including some over 100 years old), is the perfect afternoon getaway.

For the outdoor sports enthusiast, head over to the Canadian Ski Museum (200-1960 Scott Street). Founded in 1971, it’s the largest public collection of skiing memorabilia in Canada, housing over 6,000 photographs and ski equipment dating back over 150 years.

Theatre-lovers will want to check out the Centrepointe Theatre (101 Centrepointe Drive). This 1,000-seat theatre is the home of several excellent local performing groups, including Orpheus Musical Theatre Society, Savoy Society and Les Petits Ballets.

Festivals

West Ottawa is known for its many festivals – and if you’re living in a Westboro apartment building, you’ll find yourself right in the middle of the excitement. Every year, the Ottawa Folk Festival takes over Britannia Park, showcasing Canada’s folk traditions through music, dance, storytelling and crafts.

WestFest, Ottawa's newest large-scale festival, is a diverse three-day celebration that includes music, visual arts, dance, spoken word and performance art right in the heart of Westboro along Richmond Road. It features several concerts by top-level artists and is free for the general public.

Other Attractions

If physical activity is more your thing, and you’re looking at an apartment for rent in Nepean, you can’t go wrong by spending a day at the Nepean Sportsplex (1701 Woodroffe Avenue). This huge facility contains everything you need to enjoy a game of baseball, hockey, soccer, basketball, curling, football, squash, lawn bowling and much more! The Sportsplex also offers a host of wonderful programs for children and seniors.

Along the Ottawa River is the beautiful Westboro Beach, which is run by the City of Ottawa and has been a venue for public swimming since the early 1920s. The beach, which is surrounded by hiking and biking trails, also includes a spot for canoe and kayak rentals as well as a café complete with patio and kitchen.

Britannia Park is a picnicker’s paradise, with numerous tables scattered on the grass and plenty of trees to offer shade on those hot sunny days. There are two play structures and a swing set for children to play on, and it’s a great place to go swimming or windsurfing, as there is a lifeguard on duty from mid-June through to Labour Day. Also at the Park is the Ron Kolbus Lakeside Centre, a banquet hall and community centre that holds up to 300 people for special events.

The Ottawa River Parkway is a limited-access parkway that runs from Carling Avenue to the Portage Bridge near Downtown Ottawa. Packed with scenic lookouts, the Parkway is maintained by the National Capital Commission (NCC) and is complemented by over 20 km of bike paths, making it possible to ride all the way from Downtown to Kanata. On Sundays during the summer months, the Parkway is closed to all vehicle traffic for the Alcatel Sunday Bikedays – the favourite day of the week for pedestrians and cyclists alike.

Work, Play, Explore … West Ottawa Has It All!

There is always something new to discover in Ottawa’s West End – from great shopping to the great outdoors, a West Ottawa apartment has it all. See what you’re missing – take a look at Paramount’s West End apartment rentals today!

About the Author

Looking for an apartment for rent in Ottawa? Paramount Properties is Ottawa’s second largest residential property management company. With more than 4,500 Ottawa apartment rentals available in some of the city’s finest locations, we’re your #1 choice for an Ottawa apartment.