Modulo N Calculator

The modulo is the name of a mathematical operation that, for 2 numbers $ a $ and $ b $, calculates the remainder $ r $ of the Euclidean division $ a \div b $. Mathematically the modular calculus is written $$ a \equiv r \mod{b} $$

Example: A heap of $ a = 123 $ marbles divides into $ b = 10 $ heaps of $ 12 $ marbles and there remains $ r = 3 $ marbles. So $ 123 $ modulo $ 10 $ is equal to $ 3 $, or $ 123 \equiv 3 \mod{10} $

The modulo operator is sometimes noted a%b=r with the percent sign %.

Modular calculations are often imaged with a circle, like on a clock where hour calculations are done modulo 12 (or 24) for hours and modulo 60 for minutes.

Example: It is 3:00 am, in 25 hours it will be 4:00 am, is equivalent to the calculation $ 3 + 25 \equiv 4 \mod{12} $ or even (3+25)%24=4
The minute hand is $ 15 $, in $ 90 $ minutes it will be $ 45 $, because $ 15 + 90 \equiv 45 \mod{60} $

Method 1: Perform euclidean division and returns the remainder.

Example: Calculate $ A=123 $ modulo $ N=4 $, perform the Euclidean division of $ 123 / 4 = 30 \text{r} 4 $ as $ 123 = 30 \times 4 + 3 $ (the quotient is $ 30 $, and the remainder is $ 3 $). The modulo is the value of the remainder, so $ 123 \equiv 3 \pmod{4} $.

The negative modulo can be considered (rare), in this case $ 123 = 31 \times 4 - 1 $, so $ 123 \equiv -1 \pmod{4} $.

dCode uses this method that applies to both large numbers, as point numbers for A. However, N be a natural number.

Method 2: Perform the integer division and calculate the value of the difference.

Example: Calculate $ A=123 $ modulo $ N=4 $, make the division: $ 123/4 = 30.75 $. Keep the integer part $ 30 $, and multiply by $ N=4 $, $ 30 \times 4=120 $. The difference between $ 123 $ and $ 120 $ is $ 3 $, so $ 123 = 3 \pmod{4} $.

A modulo (from latin modulus) calculation can be written differently:

In Mathematics, write it using the $ \equiv $ congruence symbol and the keyword mod :

$$ 123 \equiv 3 \mod 10 $$

For computer, write the % percentage symbol, easily accessible on a keyboard:

$$ 123 \% 10 = 3 $$

In functional programming, for integers there is often the function mod() and for floating point numbers, the function fmod().

On calculators, it is often implemented with the function mod():

$$ \mod (123,10) = 3 $$

In spreadsheets like Excel, use MOD(A1;A2)

This calculus is named modular exponentiation, use the dCode page dedicated to modular exponentiation.

This calculus is named modular inverse, use the dCode page dedicated to modular inverses.

In most computation languages, the modulo operator % has the same precedence as the multiplication or division operations.