The Central Idea
Let \(a\) and \(b\) be integers with \(a\neq 0\). We say that \(a\) divides \(b\) when \(b\) can be written as an integer multiple of \(a\).
\[ \boxed{a\mid b \iff b=ak\text{ for some }k\in\mathbb Z} \]The symbol \(a\mid b\) is read “\(a\) divides \(b\).”
The key word is integer. The multiplier \(k\) must belong to \(\mathbb Z\).
A First Example
Because
\[ 42=6\cdot 7, \]and \(7\) is an integer,
\[ \boxed{6\mid 42}. \]In this relationship, \(6\) is a divisor or factor of \(42\), while \(42\) is a multiple of \(6\).
When Divisibility Fails
Consider whether \(5\) divides \(42\).
If \(5\mid 42\), there would have to be an integer \(k\) satisfying
\[ 42=5k. \]But
\[ k=\frac{42}{5}=8.4, \]which is not an integer. Therefore,
\[ \boxed{5\nmid 42}. \]The symbol \(5\nmid 42\) is read “\(5\) does not divide \(42\).”
Divisibility Is About Exact Integer Multiples
Ordinary division can produce fractions or decimals. Divisibility asks a more specific question: is the quotient an integer?
\[ 35\div 7=5 \]gives an integer quotient, so
\[ 7\mid 35. \]By contrast,
\[ 35\div 8=\frac{35}{8}, \]which is not an integer, so
\[ 8\nmid 35. \]Factor Language and Multiple Language
The statements
\[ a\mid b, \]“\(a\) is a factor of \(b\),” and “\(b\) is a multiple of \(a\)” express the same relationship.
For example,
\[ 9\mid 63 \]means all of the following:
- \(9\) divides \(63\);
- \(9\) is a factor of \(63\);
- \(63\) is divisible by \(9\);
- \(63\) is a multiple of \(9\).
Positive Divisors
A positive divisor of a positive integer is a positive integer that divides it exactly.
The positive divisors of \(12\) are
\[ \boxed{1,\ 2,\ 3,\ 4,\ 6,\ 12}. \]Each appears in a factorization of \(12\):
\[ 12=1\cdot 12=2\cdot 6=3\cdot 4. \]Integer Divisors Can Be Negative
When the full integer system is being used, negative divisors also occur. Since
\[ 12=(-3)(-4), \]we have
\[ -3\mid 12. \]Thus the integer divisors of \(12\) include
\[ \pm1,\ \pm2,\ \pm3,\ \pm4,\ \pm6,\ \pm12. \]In many introductory problems, “divisors” means positive divisors unless the problem explicitly asks for integer divisors.
Every Nonzero Integer Divides Itself
For any nonzero integer \(a\),
\[ a=a\cdot1. \]Therefore,
\[ \boxed{a\mid a}. \]For example,
\[ 17\mid17 \qquad\text{and}\qquad -8\mid-8. \]One Divides Every Integer
Every integer \(n\) can be written
\[ n=1\cdot n. \]Since \(n\) is an integer,
\[ \boxed{1\mid n} \]for every integer \(n\).
Likewise,
\[ n=(-1)(-n), \]so \(-1\) also divides every integer.
Every Nonzero Integer Divides Zero
Let \(a\neq0\). Then
\[ 0=a\cdot0. \]Since \(0\) is an integer,
\[ \boxed{a\mid0}. \]For example,
\[ 5\mid0,\qquad -12\mid0,\qquad 103\mid0. \]Why Zero Is Not Used as a Divisor Here
This lesson uses the standard introductory convention that a divisor is nonzero. Division by zero is undefined, so expressions such as
\[ \frac{b}{0} \]are not used to test divisibility.
Accordingly, when we write \(a\mid b\) in this lesson, we assume \(a\neq0\).
Divisibility and Remainders
When an integer \(b\) is divided by a positive integer \(a\), the result can be written
\[ b=aq+r, \]where \(q\) is an integer quotient and \(r\) is the remainder.
Divisibility occurs exactly when
\[ \boxed{r=0}. \]For example,
\[ 53=7(7)+4, \]so \(7\nmid53\). But
\[ 56=7(8)+0, \]so \(7\mid56\).
Multiples Form a Pattern
The integer multiples of \(5\) are obtained by multiplying \(5\) by every integer:
\[ \ldots,-20,-15,-10,-5,0,5,10,15,20,\ldots \]Every number in this list is divisible by \(5\).
More generally, the multiples of \(a\) are the numbers
\[ \boxed{ak,\qquad k\in\mathbb Z}. \]Divisibility on the Number Line
Multiples of a positive integer \(a\) appear at regularly spaced positions on the number line.
For \(a=4\), the multiples are
\[ \ldots,-12,-8,-4,0,4,8,12,\ldots \]The constant spacing of \(4\) reflects repeated addition or subtraction of the divisor.
Signs Do Not Change the Basic Relationship
If \(a\mid b\), then \(-a\mid b\) as well.
Suppose
\[ b=ak. \]Then
\[ b=(-a)(-k), \]and \(-k\) is still an integer.
Therefore,
\[ \boxed{a\mid b\implies -a\mid b}. \]If \(a\mid b\), Then \(a\mid -b\)
If
\[ b=ak, \]then
\[ -b=a(-k). \]Thus
\[ \boxed{a\mid b\implies a\mid -b}. \]Divisibility therefore behaves naturally with respect to signs.
Divisibility Is Transitive
If \(a\mid b\) and \(b\mid c\), then \(a\mid c\).
To see why, write
\[ b=am \]and
\[ c=bn \]for integers \(m\) and \(n\). Substituting gives
\[ c=(am)n=a(mn). \]Since \(mn\) is an integer,
\[ \boxed{a\mid c}. \]Example of Transitivity
We know
\[ 3\mid12 \]and
\[ 12\mid60. \]Therefore,
\[ \boxed{3\mid60}. \]The direct factorization confirms it:
\[ 60=3\cdot20. \]Divisors Survive Integer Scaling
If \(a\mid b\), then \(a\) divides every integer multiple of \(b\).
If
\[ b=ak, \]then for any integer \(m\),
\[ mb=a(mk). \]Hence,
\[ \boxed{a\mid b\implies a\mid mb}. \]Divisibility of Sums
If \(a\mid b\) and \(a\mid c\), then \(a\) divides their sum.
Write
\[ b=am,\qquad c=an. \]Then
\[ b+c=am+an=a(m+n). \]Because \(m+n\) is an integer,
\[ \boxed{a\mid(b+c)}. \]Divisibility of Differences
Under the same assumptions,
\[ b-c=am-an=a(m-n). \]Therefore,
\[ \boxed{a\mid b\text{ and }a\mid c\implies a\mid(b-c)}. \]This is one of the most useful elementary tools in number theory.
Integer Linear Combinations
The sum and difference rules extend further. If
\[ a\mid b \qquad\text{and}\qquad a\mid c, \]then for any integers \(m\) and \(n\),
\[ \boxed{a\mid(mb+nc)}. \]The expression \(mb+nc\) is called an integer linear combination of \(b\) and \(c\).
Example: Using a Linear Combination
Because
\[ 7\mid35 \qquad\text{and}\qquad 7\mid21, \]\(7\) must divide any integer linear combination of \(35\) and \(21\).
For example,
\[ 3(35)-2(21)=105-42=63. \]Therefore,
\[ \boxed{7\mid63}. \]A Useful Cancellation Property
Suppose \(c\neq0\) and
\[ ac\mid bc. \]Then there is an integer \(k\) such that
\[ bc=(ac)k. \]Cancelling the nonzero factor \(c\) gives
\[ b=ak. \]Therefore,
\[ \boxed{ac\mid bc\implies a\mid b}. \]A Bound on Positive Divisors
Suppose \(a\) and \(b\) are positive integers and \(a\mid b\). Then
\[ b=ak \]for some positive integer \(k\). Since \(k\ge1\),
\[ b\ge a. \]Thus a positive divisor of a positive number cannot be larger than the number itself:
\[ \boxed{a\mid b,\ a>0,\ b>0\implies a\le b}. \]Divisibility and Parity
An integer is even exactly when it is divisible by \(2\).
\[ \boxed{2\mid n\iff n=2k\text{ for some }k\in\mathbb Z} \]An integer that is not divisible by \(2\) is odd and can be written
\[ n=2k+1 \]for some integer \(k\).
Basic Divisibility Tests
In base ten, the decimal digits of an integer reveal several common divisibility patterns.
- A number is divisible by \(2\) when its final digit is even.
- A number is divisible by \(5\) when its final digit is \(0\) or \(5\).
- A number is divisible by \(10\) when its final digit is \(0\).
- A number is divisible by \(3\) when the sum of its digits is divisible by \(3\).
- A number is divisible by \(9\) when the sum of its digits is divisible by \(9\).
These tests provide efficient recognition rules, but the definition \(a\mid b\iff b=ak\) remains the underlying meaning.
Divisibility by \(4\) and \(8\)
A base-ten integer is divisible by \(4\) when the number formed by its final two digits is divisible by \(4\).
For example, \(7316\) is divisible by \(4\) because
\[ 16=4\cdot4. \]Similarly, a base-ten integer is divisible by \(8\) when the number formed by its final three digits is divisible by \(8\).
Thus \(18{,}632\) is divisible by \(8\) because
\[ 632=8\cdot79. \]Divisibility by \(6\)
Since
\[ 6=2\cdot3, \]and \(2\) and \(3\) share no common factor greater than \(1\), an integer is divisible by \(6\) exactly when it is divisible by both \(2\) and \(3\).
For example, \(234\) is even and its digit sum is
\[ 2+3+4=9. \]Therefore,
\[ \boxed{6\mid234}. \]Divisibility by \(11\)
A useful base-ten test for \(11\) uses an alternating sum of digits. If the alternating sum is divisible by \(11\), then the original number is divisible by \(11\).
For \(2728\),
\[ 2-7+2-8=-11. \]Since \(-11\) is divisible by \(11\),
\[ \boxed{11\mid2728}. \]Why Digit Tests Work
Divisibility tests are not isolated tricks. They arise from the structure of base-ten place value.
For example, powers of \(10\) leave predictable remainders when divided by \(3\), \(9\), and \(11\). That allows the digits of a number to replace the full number in certain divisibility questions.
Later work with congruence and modular arithmetic will make this reasoning systematic.
Divisibility Is Not Symmetric
From
\[ 4\mid20, \]we cannot conclude
\[ 20\mid4. \]In fact,
\[ \frac{4}{20}=\frac15 \]is not an integer.
Therefore divisibility is not a symmetric relationship.
Divisibility Does Not Mean “Smaller Than”
The statement \(a\mid b\) is not simply a comparison of size.
For example,
\[ 7<20, \]but
\[ 7\nmid20. \]Divisibility concerns multiplicative structure, not merely numerical order.
Divisibility Claims Can Be Proved Constructively
To prove
\[ a\mid b, \]it is enough to exhibit an integer \(k\) such that
\[ b=ak. \]For example, to prove \(13\mid247\), observe that
\[ 247=13\cdot19. \]The integer \(19\) is the required witness.
Showing That a Number Does Not Divide Another
To show \(a\nmid b\), one common method is to divide and show that a nonzero remainder remains.
For example,
\[ 100=9(11)+1. \]Since the remainder is \(1\),
\[ \boxed{9\nmid100}. \]Variable Expressions and Divisibility
Divisibility language also applies to expressions known to be integers.
For every integer \(n\),
\[ n(n+1) \]is divisible by \(2\).
The reason is that two consecutive integers always contain one even integer. Therefore,
\[ \boxed{2\mid n(n+1)}. \]Example: A Divisibility Identity
Show that \(3\) divides
\[ n^3-n \]for every integer \(n\).
Factor:
\[ n^3-n=n(n-1)(n+1). \]These are three consecutive integers. Among any three consecutive integers, one is divisible by \(3\). Hence their product is divisible by \(3\):
\[ \boxed{3\mid(n^3-n)}. \]Divisibility and Factor Pairs
If \(a\mid b\), then \(b=ak\) for an integer \(k\). Thus \(a\) and \(k\) form a factor pair for \(b\).
For \(36\), the positive factor pairs are
\[ 1\cdot36,\qquad2\cdot18,\qquad3\cdot12,\qquad4\cdot9,\qquad6\cdot6. \]Listing factor pairs is one practical way to identify all positive divisors of a moderately sized integer.
Checking Divisors Efficiently
When searching for positive divisors of a positive integer \(n\), factor pairs come in matched values.
If a factor \(d\) exceeds \(\sqrt n\), its paired factor \(n/d\) is less than \(\sqrt n\).
Therefore, to find all factor pairs, it is enough to test possible divisors up through
\[ \boxed{\sqrt n}. \]This observation will become useful when distinguishing prime numbers from composite numbers.
Example: Finding the Positive Divisors of \(40\)
Since
\[ \sqrt{40}\approx6.32, \]test positive integers from \(1\) through \(6\).
The successful factor pairs are
\[ 1\cdot40,\qquad2\cdot20,\qquad4\cdot10,\qquad5\cdot8. \]Thus the positive divisors are
\[ \boxed{1,\ 2,\ 4,\ 5,\ 8,\ 10,\ 20,\ 40}. \]Common Error: Reading the Symbol Backward
The statement
\[ 4\mid28 \]means “\(4\) divides \(28\),” not “\(28\) divides \(4\).”
A useful check is to ask whether the right-hand number can be written as the left-hand number times an integer:
\[ 28=4\cdot7. \]Common Error: Accepting a Noninteger Quotient
The fact that ordinary division produces a number does not establish divisibility.
For example,
\[ \frac{15}{4}=3.75. \]Because the quotient is not an integer,
\[ \boxed{4\nmid15}. \]Common Error: Confusing Factor with Multiple
If
\[ 8\mid56, \]then \(8\) is a factor of \(56\), while \(56\) is a multiple of \(8\).
Reversing those roles changes the meaning.
Common Error: Assuming Divisibility from Two Separate Facts
Suppose \(2\mid n\) and \(3\mid n\). Then \(6\mid n\), because \(2\) and \(3\) are relatively prime.
But this pattern cannot be copied blindly when the divisors share factors. For example,
\[ 4\mid12 \qquad\text{and}\qquad 6\mid12, \]yet
\[ 24\nmid12. \]Combining divisibility conditions requires attention to shared factors. Later lessons on greatest common divisors and least common multiples will formalize this issue.
A Reliable Procedure for Testing \(a\mid b\)
- Confirm that the proposed divisor \(a\) is nonzero.
- Compute or reason about \(b/a\).
- Determine whether the quotient is an integer.
- Equivalently, divide \(b\) by \(a\) and inspect the remainder.
- If the quotient is an integer, write \(b=ak\) to verify the claim.
- If a nonzero remainder remains, conclude \(a\nmid b\).
A Reliable Procedure for Proving a General Divisibility Statement
- Translate the claim into divisibility notation.
- Use known hypotheses such as \(a\mid b\) to write \(b=ak\).
- Substitute these forms into the expression being studied.
- Factor out the proposed divisor.
- Verify that the remaining factor is an integer.
- Conclude divisibility from the definition.
Example: Proving a Sum Property
Suppose \(5\mid a\) and \(5\mid b\). Then there exist integers \(m\) and \(n\) such that
\[ a=5m,\qquad b=5n. \]Therefore,
\[ a+b=5m+5n=5(m+n). \]Since \(m+n\) is an integer,
\[ \boxed{5\mid(a+b)}. \]Example: Proving a Difference Property
Suppose \(12\mid x\) and \(12\mid y\). Then
\[ x=12m,\qquad y=12n \]for integers \(m\) and \(n\). Hence
\[ x-y=12m-12n=12(m-n). \]Therefore,
\[ \boxed{12\mid(x-y)}. \]Example: Divisibility from Algebraic Structure
Determine whether
\[ 8\mid(24n+40) \]for every integer \(n\).
Factor:
\[ 24n+40=8(3n+5). \]Since \(3n+5\) is an integer whenever \(n\) is an integer,
\[ \boxed{8\mid(24n+40)}. \]Divisibility as a Structural Tool
Divisibility converts questions about arithmetic into statements about structure.
Instead of asking only for a quotient, number theory asks questions such as:
- which integers divide a given number;
- which numbers share common divisors;
- which integers have only trivial positive divisors;
- which remainders are possible under repeated division;
- which integer equations can have solutions.
These questions lead directly into the remaining topics of this unit.
Practice 1: Read the Notation
State whether each claim is true or false.
\[ 7\mid49,\qquad 8\mid54,\qquad -5\mid35,\qquad 13\mid0. \]Practice 2: Write a Witness
Show that
\[ 12\mid156 \]by writing \(156=12k\) for an integer \(k\).
Practice 3: Use a Remainder
Determine whether
\[ 17\mid250. \]Express \(250\) in the form \(17q+r\).
Practice 4: List Positive Divisors
List all positive divisors of
\[ 48. \]Practice 5: Use a Divisibility Test
Determine whether
\[ 3\mid 58{,}746 \]without performing long division.
Practice 6: Divisibility by \(8\)
Determine whether
\[ 8\mid 91{,}432. \]Practice 7: Apply Transitivity
Suppose
\[ 6\mid a \qquad\text{and}\qquad a\mid b. \]What can you conclude about \(6\) and \(b\)?
Practice 8: Use a Linear Combination
Suppose
\[ 11\mid x \qquad\text{and}\qquad 11\mid y. \]Show that
\[ 11\mid(4x-3y). \]Practice 9: General Divisibility
Show that \(4\) divides
\[ n(n+1)(n+2)(n+3) \]for every integer \(n\).
Practice 10: Identify the Error
A student says:
“Because \(4\mid20\), it follows that \(20\mid4\).”
Explain the error.
Practice Answers
Practice 1
The claims are
\[ \boxed{\text{true},\ \text{false},\ \text{true},\ \text{true}}. \]In particular,
\[ 49=7(7),\qquad 54=8(6)+6,\qquad 35=(-5)(-7),\qquad 0=13(0). \]Practice 2
\[ 156=12(13). \]Therefore,
\[ \boxed{12\mid156}. \]Practice 3
\[ 250=17(14)+12. \]The remainder is nonzero, so
\[ \boxed{17\nmid250}. \]Practice 4
The positive factor pairs are
\[ 1\cdot48,\quad 2\cdot24,\quad 3\cdot16,\quad 4\cdot12,\quad 6\cdot8. \]Therefore, the positive divisors are
\[ \boxed{1,\ 2,\ 3,\ 4,\ 6,\ 8,\ 12,\ 16,\ 24,\ 48}. \]Practice 5
Add the digits:
\[ 5+8+7+4+6=30. \]Since \(3\mid30\),
\[ \boxed{3\mid58{,}746}. \]Practice 6
Check the final three digits:
\[ 432=8(54). \]Therefore,
\[ \boxed{8\mid91{,}432}. \]Practice 7
Divisibility is transitive. Since
\[ 6\mid a \qquad\text{and}\qquad a\mid b, \]we conclude
\[ \boxed{6\mid b}. \]Practice 8
Write
\[ x=11m,\qquad y=11n \]for integers \(m,n\). Then
\[ 4x-3y=44m-33n=11(4m-3n). \]Since \(4m-3n\) is an integer,
\[ \boxed{11\mid(4x-3y)}. \]Practice 9
The product contains four consecutive integers. Among four consecutive integers, one is divisible by \(4\), or two separate factors are even. Either way, the product contains at least two factors of \(2\).
Therefore,
\[ \boxed{4\mid n(n+1)(n+2)(n+3)}. \]Practice 10
Divisibility is not symmetric. The fact that
\[ 20=4(5) \]proves \(4\mid20\). But there is no integer \(k\) such that
\[ 4=20k. \]Therefore,
\[ \boxed{20\nmid4}. \]What to Remember
The defining relationship is
\[ \boxed{a\mid b\iff b=ak\text{ for some }k\in\mathbb Z}. \]- Divisibility means exact integer multiplication.
- \(a\mid b\) means \(a\) is a divisor of \(b\) and \(b\) is a multiple of \(a\).
- A zero remainder is equivalent to divisibility.
- Divisibility is transitive but not symmetric.
- If a number divides two integers, it divides every integer linear combination of them.
- Divisibility tests provide efficient base-ten shortcuts for common divisors.
- Factor pairs help organize the positive divisors of an integer.
The most important habit is to return to the definition: identify the integer multiplier that makes the divisibility claim true.
Where We Go Next
Divisibility lets us classify integers according to their factors. The next lesson, Prime and Composite Numbers, studies integers whose positive divisor structure is either minimal or decomposable, establishing the vocabulary needed for factorization and later work with greatest common divisors, congruences, and arithmetic structure.