means that 
and 
differ by a multiple of 
. Or, if you prefer, that 
is a multiple of 
.
In general, 
means that 
and 
differ by a multiple of 
, or that 
is a multiple of 
.
It's explained in detail in the Wikipedia article on "modular equivalence". The 
symbol itself is pronounced "is equivalent to".
means that and differ by a multiple of . Or, if you prefer, that is a multiple of .
In general, means that and differ by a multiple of , or that is a multiple of .
It's explained in detail in the Wikipedia article on "modular equivalence". The symbol itself is pronounced "is equivalent to".
Wilson's Theorem:
It means "is congruent to" (modulo n): that is, 
And that simply means that 
divides 
.
means that and differ by a multiple of . Or, if you prefer, that is a multiple of .
In general, means that and differ by a multiple of , or that is a multiple of .
It's explained in detail in the Wikipedia article on "modular equivalence". The symbol itself is pronounced "is equivalent to".
Videos
(This explanation is only for positive numbers since it depends on the language otherwise)
Definition
The modulo operation returns the remainder or signed remainder of a division, after one number is divided by another, the latter being called the modulus of the operation. (source: wikipedia)
For instance, 9 divided by 4 equals 2 but it remains 1. Here, 9 / 4 = 2 and 9 % 4 = 1.
Image source: Wikimedia
In your example: 5 divided by 7 gives 0 but it remains 5 (5 % 7 == 5).
Calculation
The modulo operation can be calculated using this equation:
a % b = a - floor(a / b) * b
floor(a / b)represents the number of times you can divideabybfloor(a / b) * bis the amount that was successfully shared entirely- The total (
a) minus what was shared equals the remainder of the division
Applied to the last example, this gives:
5 % 7 = 5 - floor(5 / 7) * 7 = 5
Modular Arithmetic
That said, your intuition was that it could be -2 and not 5. Actually, in modular arithmetic, -2 = 5 (mod 7) because it exists k in Z such that 7k - 2 = 5.
You may not have learned modular arithmetic, but you have probably used angles and know that -90° is the same as 270° because it is modulo 360. It's similar, it wraps! So take a circle, and say that its perimeter is 7. Then you read where is 5. And if you try with 10, it should be at 3 because 10 % 7 is 3.
Two Steps Solution.
Some of the answers here are complicated for me to understand. I will try to add one more answer in an attempt to simplify the way how to look at this.
Short Answer:
Example 1:
7 % 5 = 2Each person should get one pizza slice.
Divide 7 slices on 5 people and every one of the 5 people will get one pizza slice and we will end up with 2 slices (remaining). 7 % 5 equals 2 is because 7 is larger than 5.
Example 2:
5 % 7 = 5Each person should get one pizza slice
It gives 5 because 5 is less than 7. So by definition, you cannot divide whole 5items on 7 people. So the division doesn't take place at all and you end up with the same amount you started with which is 5.
Programmatic Answer:
The process is basically to ask two questions:
Example A: (7 % 5)
(Q.1) What number to multiply 5 in order to get 7?
Two Conditions: Multiplier starts from `0`. Output result should not exceed `7`.
Let's try:
Multiplier is zero 0 so, 0 x 5 = 0
Still, we are short so we add one (+1) to multiplier.
1 so, 1 x 5 = 5
We did not get 7 yet, so we add one (+1).
2 so, 2 x 5 = 10
Now we exceeded 7. So 2 is not the correct multiplier.
Let's go back one step (where we used 1) and hold in mind the result which is5. Number 5 is the key here.
(Q.2) How much do we need to add to the 5 (the number we just got from step 1) to get 7?
We deduct the two numbers: 7-5 = 2.
So the answer for: 7 % 5 is 2;
Example B: (5 % 7)
1- What number we use to multiply 7 in order to get 5?
Two Conditions: Multiplier starts from `0`. Output result and should not exceed `5`.
Let's try:
0 so, 0 x 7 = 0
We did not get 5 yet, let's try a higher number.
1 so, 1 x 7 = 7
Oh no, we exceeded 5, let's get back to the previous step where we used 0 and got the result 0.
2- How much we need to add to 0 (the number we just got from step 1) in order to reach the value of the number on the left 5?
It's clear that the number is 5. 5-0 = 5
5 % 7 = 5
Hope that helps.
I've been trying to learn how to implement a functional quadratic sieve and have no idea where to start, but everything I come across uses the congruence symbol which I can't seem to get my head around:
a == b (mod n)
except replace the == with an equal sign with three lines in it.
I think the three most common notations are:
- $x \bmod n$
- $x \operatorname{rem} n$
- $x \% n$
The first is an abuse of notation and somewhat misleading to people learning the subject, but it is common. The others, IMO, would be readily understood if you state once at the beginning of whatever you're writing what the notation means.
For your specific application you could also use the iverson bracket:
- $ [i \text{ is odd} ]$
The Iverson bracket is basically the mathematical incarnation of the usual ways to coerce a boolean value to an integer: it gives $0$ when false and $1$ when true.
In the specific formula you write, a possibly more useful alternative is $$ 2^{[i \text{ odd} ]} = 1 + [i \text{ odd} ]$$ since this form is more amenable to doing series manipulations. In fact, the context of series manipulations is the first time I saw extensive use of the Iverson bracket.
Given the integer $x$ and the modulus $n$, it would be standard to write $$ x \mod n. $$ To be a little more precise, if $r$ is the remainder of $x$ after division by $n$ in accordance with the Division Algorithm, we have $$ x \equiv r \mod n. $$
I know the percent sign as the modulo operator in a few programming languages but in maths everyone uses "mod". Why isn't there a symbol like + for addition or minus for subtraction?
I suspect it is because "modulo" describes an equivalence class, rather than a particular number: 4 (mod 7) is technically the set of all integers n such that 7|(n-4); this set is precisely the same set as -3 (mod 7), so what should "25%7" equate to? 4? -3? 25? I don't believe it is common enough as an operation in mathematics to warrant identifying a canonical result: e.g. as the unique representative between 0 and n-1; it is more commonly used to set up the structure being worked on/in (e.g. Z_p, integers modulo p).
Many programming languages use %. http://en.wikipedia.org/wiki/Modulo_operation
But there is ambiguity on the output, and you have to be careful about what you really want.
Hello everyone,
In modular arithmetic, if we know the remainder r when dividing a by m, we write it as:
a ≡ r mod m
As I understand it, r is the result of the operation a mod m.
However, in other formulas—like in RSA encryption—we often see something like:
y ≡ x^(e) mod m
This means that y is the result of the operation x^(e) mod n.
So to me, it would feel more intuitive to write:
x^(e) ≡ y mod n
since x^(e) mod n = y, and the expression being reduced appears on the left-hand side.
The way the modular expression is written can be a little confusing at first, but both forms describe the same relationship.