Another way to think of congruence modulo, is to say that integers a and b congruent modulo n if their difference is a multiple of n. So, what have we learned? If the remainders are equal to each other, then they are congruent! And the following theorem highlights this very idea:. In our lesson, we will work through several proofs for these essential theorems, as they are pivotal in our understanding of equivalence relations future lesson and allow us to define arithmetic operations.
The definition of addition and multiplication modulo follows the same properties of ordinary addition and multiplication of algebra. Together we will work through countless examples of modular arithmetic and the importance of the remainder and congruence modulus and arithmetic operations to ensure mastery and understanding of this fascinating topic. Get access to all the courses and over HD videos with your subscription.
Get My Subscription Now. Please click here if you are not redirected within a few seconds. So we can use modulo to figure out whether numbers are consistent, without knowing what they are! A contradication, good fellows! The modular properties apply to integers, so what we can say is that b cannot be an integer.
Playing with numbers has very important uses in cryptography. Geeks love to use technical words in regular contexts. Happy math! Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. Odd, Even and Threeven Shortly after discovering whole numbers 1, 2, 3, 4, 5… we realized they fall into two groups: Even: divisible by 2 0, 2, 4, Where will the big hand be in 25 hours?
Uses Of Modular Arithmetic Now the fun part — why is modular arithmetic useful? This is a bit more involved than a plain modulo operator, but the principle is the same. Putting Items In Random Groups Suppose you have people who bought movie tickets, with a confirmation number.
Picking A Random Item I use the modulo in real life. Know its limits: it applies to integers. Cryptography Playing with numbers has very important uses in cryptography. Plain English Geeks love to use technical words in regular contexts. In general, I see a few general use cases: Range reducer: take an input, mod N, and you have a number from 0 to N So far, we've only talked about notation.
Now let's do some maths, and see how congruences what we've described above can make things a bit clearer. Here are some useful properties. We can add congruences. Why is this? You can prove this in the same way that we used above for addition.
Division is a bit more tricky: you have to be really careful. Here's an example of why. In general, it's best not to divide congruences; instead, think about what they really mean rather than using the shorthand and work from there.
Let's think about an example.
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