7.4.2. Congruence with Respect to Modulo

Now, let us define what we call as congruence with respect to modulo. So if you have a modulus N, then a will be; then we say that a is congruent to b modulo N if the remainder that we obtained by dividing a by N is same as, the remainder that we obtained by dividing b by N.

In this introduction, we start by defining congruence with respect to modulo, using a positive integer known as the modulus N. If two numbers, a and b, when divided by N, have the same remainder, we say that 'a is congruent to b modulo N'. This is a way of stating that both numbers yield the same result when divided by N. The notation for this is written as a ≡ b (mod N).

Imagine you have two different sets of colored balls, and you want to check if they can be evenly divided among a fixed number of friends (represented by N). If both sets leave the same number of balls unshared, we say there's a congruence between the two groups of balls with respect to that number of friends.

That means if a modulo N and b modulo N are the same then I will say that a is congruent to b modulo N and the notation that we use is the following: [a ≡ b mod N]. So in some sense, you can imagine that I am trying to say that a and b are equivalent, in the sense that they give you the same remainder on getting divided by N.

Here we emphasize the importance of the modulo operation in determining congruence. When we take 'a modulo N' and 'b modulo N', we essentially extract the remainders of a and b when divided by N. If these remainders are identical, we declare both integers a and b as congruent modulo N, indicating they behave similarly under division by N.

Think of it like two clocks showing the same hour, even though one is in New York, and the other is in London. No matter how far apart they are, if both clocks point to 3 PM, they're congruent with respect to the 12-hour cycle. The emphasis here is on the equivalence of the two different locations based on their respective modulo N (the hours on a clock).

It is very easy to verify that if a is congruent to b modulo N, then that is possible if and only if a - b is completely divisible by N.

This chunk explains an important property of congruence: if two integers are congruent modulo N, their difference must be divisible by N. This means that if you subtract one number from the other and the result can be evenly divided by N (without any remainder), it confirms that both numbers share the same remainder when divided by N.

Imagine you have two jars of candies. If one jar has 10 candies and the other jar has 50 candies, the difference is 40. If we divide this difference by the number of friends (N = 10), the result is 4, which means these two amounts of candies will behave the same way when shared among those friends: both jars can keep being divided into portions of 10 candies. This illustrates that their difference being divisible by 10 confirms their congruence.

So there are some interesting arithmetic rules which your modular arithmetic follows. So imagine that a modulo N is a’ and b modulo N is b’. Then a + b modulo N will be the same as a’ + b’ modulo N, a - b modulo N will be same as a’ - b’ modulo N so on.

This chunk introduces some fundamental arithmetic rules associated with congruence. If we take two numbers a and b and find their respective remainders when divided by N, we can add, subtract, or multiply these results, and it will yield the same result as performing these operations on a and b first and then taking the modulo N. This property allows us to simplify calculations significantly.

Picture you're baking cookies, and you can either mix all the ingredients and then bake them (direct calculation) or you can just measure what you need (a’ for flour, b’ for sugar) before mixing. Both methods will lead to the same final batch of cookies, illustrating how you can work with smaller numbers and achieve the same results.

Now, we have seen the rules for addition, subtraction, multiplication, what about division? So, imagine a modulo N is a’ and b modulo N is b’ of course, a’ and b’ are in the range 0 to N - 1. Now, what can I say about a over b modulo N and a’ over b’ modulo N? Can I say that these two expressions will be the same? Well, the answer is no because at the first place the value a over b modulo N may not be well defined.

This section takes us into the complexities of division in modular arithmetic. Unlike addition and multiplication, division in the modular context isn't straightforward. The division of two numbers a and b can often lead to non-integers, which means it doesn't align well with the definition of modular arithmetic. Thus, we cannot directly infer that if a modulo N equals b modulo N, then a divided by b modulo N will also hold true.

Imagine you're trying to divide a pizza (a) among friends (b). If everyone must get a whole slice, how do you divide it evenly if the number of slices isn't straightforward (like having leftover crusts)? If a is a whole pizza separated into say 12 slices and b is another person who also wants more than 1 slice, but it's not evenly shared, you can see the division isn't as clear-cut as simply counting. This demonstrates why division poses unique challenges in modular arithmetic.

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