# The quotient remainderÂ theorem

## The quotient remainder theorem

When we want to

It is a simple idea that comes directly from

**prove some properties**about**modular arithmetic**we often make use of the**quotient remainder theorem**.It is a simple idea that comes directly from

**long division**.The quotient remainder theorem says:

Given

Given

**any**integer**A**, and**a positive**integer**B**, there exist**unique integers Q and R**such that**A= B * Q + R where 0 â‰¤ R < B**

We can see that this comes directly from long division. When we

If we can write a number in this form then

**divide A by B**in long division, Q is the quotient and**R is the remainder**.If we can write a number in this form then

**A mod B = R**## Examples

**A = 7**,

**B = 2**

**7**=

**2*** 3 +

**1**

**7**mod

**2**=

**1**

**A = 8**,

**B = 4**

**8**=

**4*** 2 +

**0**

**8**mod

**4**=

**0**

**A = 13**,

**B = 5**

**13**=

**5*** 2 +

**3**

**13**mod

**5**=

**3**

**A = -16**,

**B = 26**

**-16**=

**26*** -1 +

**10**

**-16**mod

**26**=

**10**