The **modulo function** represents the remainder that results from dividing two different positive numbers. For example, 7 modulo 2 is 1 because the division of 7 by 2 has a quotient of 3, leaving a remainder of 1. Although the modulo function initially seems like an esoteric mathematical function, it does have a few practical applications including identifying odd or even numbers in worksheets, databases, or complex formulas (Bluttman & Aitken, 2018); In programming, the modulo can be used to alternate values in a loop.

## Formal Definition of Modulo Function

An arbitrary positive integer, a, can always be written in the following form

a = n * q + r

Where:

- q is the quotient,
- r is the remainder, and
- n is an integer.

Within this formal definition, the modulo expression can be solved: (a mod n) = r.

## Calculating the Modulo

The basic steps are:

**Step 1**: Set up the equation. Given the expression (a mod n), set up the equation a = n * q + r.

**Step 2**: Maximize the quotient, q.

**Step 3**: Solve for the remainder, r.

**Step 4**: (a mod n) = r.

## Example Problems

Problem 1: Find (18 mod 5) using the formal definition of modulo.

Solution:

Step 1: In the context of the formal definition, a = 18 and n = 5. Therefore,

18 = 5 * q + r

Step 2: Maximize the quotient. For this equation, q=3.

Step 3: Solve for the remainder:

- 18 = 5 * 3 + r
- 18 = 15 + r
- r = 3

Therefore, (18 mod 5) = 3.

Problem 2: Find (17 mod 4) using the formal definition of modulo.

Solution: Within the formal definition of modulo, we identify that a=17 and n=4. Therefore,

17 = 4 * q + r

The maximum value for the quotient is q = 4.

Solving for the remainder,

17 = 4 * 4 + r

17 = 16 + r

r = 1

Thus, (17 mod 4) = 1.

Problem 3: Find (3 mod 1) using the formal definition of modulo.

Solution: From the formal definition of modulo, we have

3 = 1 * q + r

In this case, the maximum value of q=3.

3 = 1 * 3 + r

Solving for the remainder, r = 0. In fact, *any integer modulo 1 is always equal to zero, since q=a*.

## Special Considerations with the Modulo Function

- The modulo function only accepts positive natural numbers. You cannot evaluate the modulo of a negative number or a fraction.
- The remainder can only be calculated after the quotient is evaluated correctly. If the quotient is not at its maximum value, then your calculation for the remainder will be incorrect.

## References

Bluttman, K. & Aitken, P. Excel Formulas and Functions For Dummies. 2018..

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