In this section we investigate the properties of the operation \(\fmod\) and show how these can be applied. The properties will seem awkward at first but will turn out to be powerful tools in computations when numbers get larger. In the video in Figure 3.37 we give an overview of the properties of \(\fmod\) covered in this section.
Recall that \(a\fmod b\) is the remainder of the division of \(a\) by \(b\) (see Definition 3.21). We have established that the division algorithm (Algorithm 3.6 and Algorithm 3.14) produces the quotient and remainder of a particular division as its output values. As an alternative method we have presented (calculator) long division in Strategy 3.1.
Example3.38.Three methods for computing \(41\fmod 13\).
We compute \(41\fmod 13\) in three different ways. The number \(41 \fmod 13\) is the remainder of the division of \(41\) by \(13\text{.}\)
Method 1 the trained eye.
The largest multiple of 13 that is less than 41 is 39. The difference between 41 and 39 is 2, this is the remainder of the division of 41 by 13. Thus 41 mod 13 = 2.
We have \(41\div 13=3.153\dots\text{.}\) So the quotient from the decision of \(41\) by \(13\) is the largest integer that is less than \(3.153\dots\text{,}\) which is \(3\text{.}\) The remainder is \(41-3\cdot 13 = 41-39 = 2\text{.}\) Thus \(41 \mod 13 = 2\text{.}\)
Example3.39.Three methods for computing \(-41\fmod 13\).
We compute \(-41\fmod 13\) in three different ways. The number \(-41 \fmod 13\) is the remainder of the division of \(-41\) by \(13\text{.}\)
Method 1 the trained eye.
The smallest multiple of \(13\) that is greater than \(41\) is \(52\text{.}\) The sum of \(-41\) and \(52\) is \(11\text{,}\) this is the remainder of the division of \(-41\) by \(13\text{.}\) Thus \(-41 \fmod 13 = 11\text{.}\)
We have \(-41\div 13=-3.153\dots\text{.}\) So the quotient of the division of \(41\) by \(13\) is the largest integer that is less than \(-3.153\dots\) which is \(-4\text{.}\) The remainder is \(-41+4\cdot 13 = -41+52 = 11\text{.}\) Thus \(-41 \fmod 13 = 11\text{.}\)
An interesting pattern occurs when we compute the remainder of consecutive integers from the divisions by the same number. As illustrated in the following two examples, the remainders repeat in a periodic fashion.
The remainder of division by \(2\) is an integer that is greater than or equal to \(0\) and less than \(2\text{,}\) that is, it is either \(0\) or \(1\text{.}\) We use this to define two familiar terms.
We now investigate some properties of the operation \(\fmod\text{.}\) In particular, we are interested in the behavior of \(\fmod\) in sums and products.
\begin{align*}
a\amp =(a \fmod m)+(s\cdot m) \text{ for some integer } s, \text{ and }\\
b\amp =(b \fmod m)+(t\cdot m) \text{ for some integer } t\text{.}
\end{align*}
Using the theorem may seem more awkward right now. When calculations get more involved its value will become more apparent. The theorem also can be used to evaluate expressions when we only know the remainders.
As we know that \(a\fmod 113=29\) and \(b\fmod 113=100\) we can replace \(a\fmod 113\) by \(29\) and \(b\fmod 113\) by \(100\text{.}\) Copying what we have so far and evaluating we get:
In Checkpoint 3.51 decide whether statements about \(\fmod\) are true or false. If the statement is false give a counterexample, otherwise leave the box empty.
Counterexample: The statement is false, because for the integer \(a=\) we have \((a\cdot 5) \bmod 6 \ne \left((a \bmod 6) \cdot ( 5 \bmod 6)\right) \bmod 6\text{.}\)
Example3.53.Applying Theorem 3.46 and Theorem 3.50.
We compute \((61+(9\cdot 12))\fmod 7\text{.}\) We first evaluate the inner parenthesis and then the outer parenthesis. By Theorem 3.46 and Theorem 3.50 we have
In Checkpoint 3.55 apply properties the properties of \(\fmod\) to conduct computations modulo a given integer, when the values are only known modulo that integer.
In the following me make use of the decimal representation of numbers, to find remainders of divisions of numbers that would be to large to handle for most calculators. If you need a refresher on decimal numbers, flip ahead and read Section 11.1.