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NUMBER THEORY
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Index
Contents
Division of Integers ....................................
Confusions to watch out for .........................
Facts about division .....................................
Images and metaphors for division ..............
Congruence mod n ....................................
Properties of congruence .............................
Images and metaphors for congruence .......
Number theory is the study of the properties of the integers,
particularly concerning prime numbers. It is often one of the first courses after
calculus that math majors take. Some parts of number theory have important uses
in computing science.
It might be better to call it “integer theory”, but the name “number theory” has been
around for more than a century so I think we are stuck with it. (See cognitive dissonance.)
Division of Integers
Definition An integer k divides a nonzero integer m (written k | m ) if there is an
integer q with the property that m = q k .
Examples
 3 | 6 (“3 divides 6”) because 6 and 3 are integers, 6 is not zero, and 6 = 2 ґ 3
(so q = 2 ).
 3 does not divide 7 because there is no integer q with the property that 7 = 3q .
(It is true that 7 = 3 ґ (7 / 3) but 7 / 3 is not an integer.)
 5 does not divide 0 because the definition requires m to be nonzero. (A few
authors do not require that m be nonzero.) Notice that 0 = 0 ґ 5 but 5 does not divide
0 anyway.
Usage
You can word “ k | m ” in any of these ways:




k divides m.
k is a divisor of m.
k is a factor of m.
m is divisible by k.
Confusions to watch out for
 Don't confuse the vertical line “|”, a verb meaning “divides”, with the slanting line
“/” used in fractions. The expression “ 3 | 6 “ is a (true) sentence, but the expression
“ 3 / 6 “ is the name of a number (the number 1/ 2 ), and does not form a complete
sentence in itself. An additional source of confusion is the fact that the numbers are
reversed between the two notations: 3 | 6 because 6 / 3 is an integer.
 In the notation “ k | m ” and in the phrase “ m = q k ” in the definition, the k and
the m occur in opposite order. Thus 3 | 6 because 6 = 2 ґ 3 . When you do pattern
matching this sort of switch can really trip you up.
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 The definition of “divides“ requires that the numbers involved be integers. So it
doesn't make sense in general to talk about one real number dividing another. It is
tempting, for example, to say that 2 divides 2π, but according to the
definition given here, that statement is meaningless.
2 (quotient)
 You must take the definition literally. Using the letter q in
3 7
the definition may suggest to you that in the statement m = qk , q is
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the quotient when m is divided by k. Indeed it is. But it would be a
1 (remainder)
bad thing to think of q as a quotient. The definition does not say
“the quotient when m is divided by k must be an integer”. After all, in the terminology of
grade school, 2 is the quotient when 7 is divided by 3 (see box), but that does not
mean that 3 divides 7. Many authors use “integer quotient” as a name for q to avoid
this confusion. (See cognitive dissonance).
Don’t read things into the definition that are not there
Facts about division
To Do
Images and metaphors for division
Congruence mod n
Definition
Let k be a positive integer and m and n be any integers. Then m is congruent to n
mod k if k divides m - n .
Notation
 In number theory, the standard notations m є n (mod k ) or m є
k
n is used to
mean that m is congruent to n mod k .
 Computer scientists may write m mod k = n mod k to mean m is congruent to n
mod k . In that notation, “ m mod k ” means the remainder when m is divided by k. This
is the way MOD is used in modern computer languages. See here for the connection
between remainders and congruence.
Examples
 61 є 12(mod7) because 7 divides 61–12, which is 49. This could also be written
as 61 є 7 12 or 61(mod7) = 12(mod7) , the latter meaning that
61(mod7) = 12(mod7) = 5 .
 12 є 61(mod7) because 7 divides 12–61, which is –49.
Properties of congruence
Theorem
For a fixed positive integer k, congruence (mod k) is an equivalence relation on Z .
Proof
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 Reflexive: Must show that for all integers m, m є m (mod k ) . This is correct
because k divides m – m, which is zero (every integer divides 0).
 Symmetric: Must show that for all integers m and n, if m є n (mod k ) then
n є m (mod k ) . Rewriting, we must show that if k divides m – n, then k divides n – m.
Rewriting again using the definition of divides, we must show that if there is an integer q
for which m - n = qk then there is an integer q ' for which n - m = q ' k . To accomplish
this, let q ' = - q .
 Transitive: Must show that for all integers m, n and p, if m є n (mod k ) and
n є p (mod k ) then m є p (mod k ) . Rewriting again, we must show that if k divides m –
n and k divides n – p, then k divides m – p. Rewrite yet again: We must show that if there
are integers q and q ' for which m - n = qk and , then there is an integer q " for which
m - p = q " k . This calculation pulls a rabbit out of a hat to do this:
(q + q ')k = qk + q ' k = m - n + n - p = m - n
so we can let q " = q + q '.
Definition
Let k be a positive integer and m any integer. The remainder when m is divided by
k is just the remainder when you do long division. The most usable mathematical
definition is:
. The remainder when m is divided by k is the unique integer r satisfying
a)
0Ј r < k
b)
m = qk + r for some integer q
To make this a valid definition you must show that there is just one integer satisfying
a) and b).
Examples
 The remainder when 65 is divided by 7 is 2. This is because
a) 0 Ј 2 < 7 and b) 65 = 9 Ч7 + 2 . The picture on the right shows
the calculation using long division.
9
7 65
63
2
(quotient)
(remainder)
Theorem
Two positive integers m and n are congruent mod k if and only if
m and n leave the same remainder when divided by k.
Proof
Theorem
If m є m '(mod k ) and n є n '(mod k ) then m + n є m '+ n '(mod k ) and
mn є m ' n '(mod k )
Proof
Remarks
 This theorem means that if you have an expression involving integers, addition
and multiplication, you can freely substitute integers congruent to the integers you
replace and the expression will evaluate to an integer that, although it may be different,
will be congruent (mod k) to the original value.
 Mathematicians describe this fact as saying that “addition and multiplication are
compatible with congruence”.
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Examples
Images and metaphors for congruence
Clock