Modulo Calculator

The Modulo operation is a method to find the division operation remainder. The specifics of modulo are that it returns the remainder as a whole number.

Consider having three children. You buy a box of candies containing 20 of them. You would like to divide all the candies evenly and fairly among your children. And to eat the candy left by yourself without the need to cut it or break it. Your children are still at school. So you can determine the remainder left after division first and eat your number of candies.

This is the case when you can use the modulo operator. It can also be represented as % sign or mod. For operations with small numbers, you can perform calculations in your head. If you are working with large numbers, you will be more comfortable using a modulo calculator.

The equation can be represented as follows:

Dividend = (Quotient × Divisor) + Remainder

In our case:

• the dividend is 20 (the total amount of candies);
• the divisor is 3 (the number of children);
• the quotient is 6 (the number of candies for each child);
• the remainder is 2 (the number of candies you can take for yourself).

If you use the modulo operation, you can write it down in the following form:

x % y = r

or

x mod y = r

Where x is the dividend, y is the divisor, and r is the reminder.

In our case,

20 % 3 = 2

Calculations Without a Modulus Calculator

Let's take a specific case as an example.

Example

Wayan lives in Bali and is building a small guesthouse with six residential units. He is going to tile the bathrooms. His neighbor, Gede, who has already finished building his hotel, offers Wayan a considerable discount to buy the remaining tiles.

The neighbor counted 15 boxes in his warehouse, each containing 4 (60 × 60 cm) tiles and two tiles separately. So that is 62 tiles in total. And Gede wants to sell all the tiles at once.

Now Wayan needs to figure out how many bathrooms he will be able to pave with these tiles. And how many tiles will possibly remain unused?

How to find modulus manually without any modulus operator calculator?

Wayan measured the size of a standard bathroom in his guesthouse and realized he needed about 14 tiles per room.

Let's do the manual calculations!

1. Decide on a starting number or dividend. In our case, that's 62, or the number of tiles a neighbor offers.
2. Determine the divisor. This is 14—the average number of tiles for a standard bathroom.
3. Divide the dividend by the divisor and round the result to a whole number. 62 / 14 = 4.428571428571429 or 4. So Wayan can use the tiles for four bathrooms.
4. Multiply the rounded result of division by the divisor. And that is 4 × 14 = 56. This will be the number of tiles for four rooms.
5. Subtract this multiplication result from the original dividend. That is 62 - 56 = 6. This leaves Wayan with six extra tiles.

In a simplified and shortened way, we can write this operation as:

62 % 14 = 6

or

62 mod 14 = 6

Wayan decided this is a good option because he should take a tile reserve of about 10% for the tiling work in case of trimming or misunderstandings. And he will buy the tiles for the other two bathrooms at a local building store.

A mod calculator could provide this result in just seconds.

Clock Demonstration of the Modulo Principle

A type of mathematics called "modular arithmetic" deals with cyclic structures. The easiest way to represent this is a dial with a cycle of 12. For a mathematician, the dial has mod 12.

If you want to see if you can divide 251 hours by days without a remainder, you can apply the operation

251 mod 24

The result is 11, so the answer is no! We can only answer "yes" if the result is 0.

Example

Daniel wants to take a bus from Atlanta to Miami. It leaves at 1 p.m., and the road takes 15 hours. What time will it be when he arrives? That would be

1 + 15 mod 12

which is 4. In his case, it will be 4 a.m.

Using Modulo

Determining Even and Odd Numbers

One of the most basic uses of the modulus operator is to determine whether a number is even or odd. This is possible because x % 2 always returns either 0 or 1. Even numbers always return 0 because they are evenly divided by 2, while odd numbers always return a remainder of 1.

The most common case of using modulo in programming is when you print a table in your application and want to alternate colors in the rows. You might wish to color them light blue and light gray, so you check modulo to see whether you're on an even or odd row.

Unit conversion

Unit conversion is a typical example of the practical use of the modulo operation. It is usually used when we want to convert a smaller unit, such as minutes, inches, or centimeters, to a larger unit, such as hours, miles, or kilometers. Decimal or fractional numbers are not always helpful in such situations.

For example, if we want to know the number of hours in 373 minutes, a result expressed as 6 hours and 13 minutes might be more valuable than 6.2166666666666666667 hours.

Standard division (with rounding to the nearest whole number) determines the number of hours, and the modulo operation is used to account for the remaining minutes. Whether dealing with time, distance, pressure, energy, or data storage, you can use this general approach to convert units.

Determining a Leap Year

Another example of using the modulo operator is to see if a year is a leap year.

A leap year is a calendar year containing an extra day in the solar calendar. The extra day in a leap year is February 29.

On January 1, 45 B.C., the Roman dictator Gaius Julius Caesar introduced the calendar developed in Rome by Alexandrian astronomers. The calendar was based on the calculation that an astronomical year is approximately 365.25 days (365 days and 6 hours). This calendar was called the Julian calendar.

To equalize the six-hour shift, Caesar introduced a leap year. For three consecutive years, there were 365 days in a year. And each year, multiple of four, one extra day was added in February.

However, as time passed, it turned out that this rule alone was not enough.

The average tropical year (the time between the two vernal equinoxes) is more accurately about 365 days and 5 hours and 49 minutes. The difference between the average year and the Julian calendar year (365 days and 6 hours) was about 11 minutes. So in about 128 years, those 11 minutes could add up to a whole extra day.

To compensate for the accumulated errors and to avoid a similar shift in the future, Pope Gregory XIII reformed the calendar in 1582. He added additional rules for leap years. Leap years were still a multiple of four, but exceptions were made for those years which were a multiple of 100. Such years were leap years only if they were also divisible by 400.

The rules for determining the leap year became as follows:

• A year which number is a multiple of 400 is a leap year.
• The rest of the years which total is a multiple of 100 are not leap years (for example, the years 1700, 1800, 1900, 2100, 2200, 2300);
• The rest of the years, which number is a multiple of 4, are leap years.
• All other years are not leap.

Thus, the years 1700, 1800, and 1900 are not leap years, as they are a multiple of 100 and not a multiple of 400. Years 1600 and 2000 are leap years, as they are a multiple of 400.

Let's return to our problem.

We know that:

• If a year number mod 4 = 0, and a year number mod 100 ≠ 0 then it is a leap year.
• If a year number mod 400 = 0, then it is a leap year
• In any other case, it is not a leap year.

With a simple Python script, you can tell if a year is a leap year or not. It will look like this:

year = int(input('Enter year: '))

if (year%4 == 0 and year%100 != 0) or (year%400 == 0) :

	print(year, "is a leap year.")

else:

	print(year, "is not a leap year.")

Popular applications of the modulo operator in programming include:

• determining whether something is even or odd;
• performing a specific operation on each Nth item in a list;
• restricting number to range;
• rotating through limited options (circular array);
• reversing a number;
• converting linear data to a matrix;
• determining if arrays are rotated versions of each other;
• pagination.

Random number generators

Modulo matching is often used in computer hardware and telecommunications equipment to create control numbers and get random numbers in a limited range, such as a congruent random number generator. Derrick Henry Lemer proposed the linear congruent method in 1949.

The linear congruent method works according to the formula:

$$X_{n+1} = (a × X_n + c)\mod m$$

Where:

• m is the modulo,
• a is the multiplier,
• c is the increment, and
• X₀ is the initial value.

For example, for m = 11, X₀ = 9, a = 9, c = 9, we get the following series of random numbers:

9, 2, 5, 10, 0, 9, 2, 5, 10, 0, 9

Cryptography

Cryptographers love modulo. Because when used with really big numbers, you can create with modulo something known as "one-way functions." These special functions make it easy to calculate something in one direction but not the opposite direction.

If 9 is the result of squaring, you can quickly determine that the input was 3. You can imagine the whole process in front of you from beginning to end. If I tell you that 9 is the result of mod 29, it's harder to figure out what's in the input.

Cryptographers like this idea because they can use division with the remainder to generate giant prime numbers to create cryptographic keys.

Conclusion

Whether you are trying to distribute objects evenly in a storage box, knowing whether a number is divisible by another number, or simply trying to calculate time, modulo is always there. In all these cases, the remainder is as crucial as the quotient in the division operation.

Sometimes the problem at hand is straightforward and intuitive. However, it is always better to use the modulus calculator online to find the solution when things are complicated.