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Integers and Modern Applications for the Uninitiated

Sebastian Pauli

Section 4.1 Divisibility

We begin by introducing terminology.
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Definition 4.1.

Suppose that for integers \(a\) and \(b\text{,}\) there is an integer \(q\) such that \(a = b \cdot q\text{.}\) Then \(b\) divides \(a\text{.}\)
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By Definition 4.1 if \(b\) divides \(a\text{,}\) then \(a = b \cdot q\) for some integer \(q\text{.}\) Then we have \(a = b \cdot q + 0\) so that in particular \(a \fmod b = 0\text{.}\) If \(b\) does not divide \(a\text{,}\) then \(a \fmod b\ne 0\text{.}\) It follows immediately that if \(b\) divides \(a\) then \(b\le a\text{.}\)
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Problem 4.2. Determine divisibility.

For the given values of \(a\) and \(b\text{,}\) determine whether or not \(b\) divides \(a\text{.}\) If \(b\) divides \(a\text{,}\) determine the integer \(q\) such that \(a = b \cdot q\text{.}\)
  1. \(a=30\) and \(b=10\)
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  2. \(a=2\) and \(b=46\)
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  3. \(a=29\) and \(b=4\)
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Solution.

In each case we consider the remainder \(a\fmod b\) of the division \(a\) and \(b\text{.}\)
  1. We compute \(30 \fmod 10 = 0\text{.}\) So \(10\) divides \(30\text{.}\) Furthermore, we have that \(30\fdiv 10=3\) so \(30=10\cdot 3\text{.}\)
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  2. We compute \(2 \fmod 46 = 2\ne 0\text{.}\) So \(46\) does not divide \(2\text{.}\) (Be careful not to mix up \(a\) and \(b\) during the division or the conclusion. The order matters. It turns out that \(2\) does divide \(46\) since \(46 = 2\cdot 23\text{.}\))
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  3. We compute \(29 \fmod 4 = 1\ne 0\text{.}\) So \(4\) does not divide \(29\text{.}\)
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Use the method from the solution of Problem 4.2 to determine divisibility in Checkpoint 4.3.
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Checkpoint 4.3. Determine divisibility.

Compute the remainder and complete the statement about divisibility
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Because \(34 \bmod 15\)= we have that \(15\)
\(34\text{.}\)
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Because \(51 \bmod 7\)= we have that \(7\)
\(51\text{.}\)
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Because \(9 \bmod 3\)= we have that \(3\)
\(9\text{.}\)
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Because \(20 \bmod 4\)= we have that \(4\)
\(20\text{.}\)
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Because \(60 \bmod 17\)= we have that \(17\)
\(60\text{.}\)
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Because \(18 \bmod 16\)= we have that \(16\)
\(18\text{.}\)
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There are several other formulations for \(b\) divides \(a\text{.}\)
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Definition 4.4.

Let \(a\) and \(b\) be integers. If \(b\) divides \(a\) we also say:
  1. \(a\) is divisible by \(b\)
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  2. \(a\) is a multiple of \(b\)
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  3. \(b\) is a divisor of \(a\)
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  4. \(b\) is a factor of \(a\)
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In Checkpoint 4.5 we give several statements about divisibility formulated in various ways. Decide which statements are true and which statements are false.
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Checkpoint 4.5. Decide which statements are correct.

Enter T or F depending on whether the statement is a true proposition or not. (You must enter T or F -- True and False will not work.)
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  1. 20 is a factor of 220
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  2. 220 divides 20
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  3. 20 divides 220
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  4. 220 is a divisor of 20
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  5. 20 is a factor of 220
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  6. 20 is a divisor of 220
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If a number divides two other numbers, it divides their sum.
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Proof.

As \(b\) divides \(a\text{,}\) there is an integer \(q\) such that \(a=b\cdot q\text{.}\) As \(b\) divides \(c\text{,}\) there is an integer \(s\) such that \(c=b\cdot s\text{.}\) With substitution and the distributive property we obtain
\begin{equation*} a+c=(b\cdot q)+(b\cdot s)=b\cdot(q+s)\text{.} \end{equation*}
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Thus \(a+c\) is a multiple of \(b\) which means that \(b\) divides \(a+c\text{.}\)
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Example 4.7.

Let \(b:=10\) and \(a:=100\) and \(c:=1000\text{.}\) Then \(b\) divides \(a\) and \(b\) divides \(c\text{.}\) Also \(b\) divides \(a+c=1100\text{.}\)
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