PROFESSOR: The
idea of congruence
was introduced to
the world by Gauss
in the early 18th century.
You've heard of him
before, I think.
He's responsible for some
work on magnetism also.
And it turns out that this
idea, after several centuries,
remains an active field of
application and research.
And in particular,
in computer science
it's used significantly
in crypto,
which is what we're going to be
leading up to now in this unit.
It's plays a role
in hashing, which
is a key method for
managing data in memory.
But we are not going to
go into that application.
Anyway, the definition of
congruence is real simple.
Congruence is a relation
between two numbers, a and b.
It's determined by
another parameter
n, where n is considered
to be greater than one.
All of these, as
usual, are integers.
And the definition
is simply that a
is congruent to b mod n if n
divides a minus b or a minus
b is a multiple of n.
So that's a key
definition to remember.
There's other ways to define it.
We'll see very shortly
an equivalent formulation
that could equally well have
been used as a definition.
But this is a standard one.
A is equivalent to b means that
a minus b is a multiple of n.
Well let's just practice.
30 is equivalent to 12 mod
9 because 30 minus 12 is 18,
and 9 divides 18.
OK.
An immediate
application is that does
this number with a lot
of 6's is ending in a 3
is equivalent to
788253 modulo 10.
Now why is that?
Well, there's a
very simple reason.
If you think about subtracting
the 6 number ending
in 3 from the 7 number ending
in 3, what you can immediately
see without doing much of
any of the subtraction--
just do the low order digits--
when you subtract these,
you're going to get a
number that ends in 0.
Which means that
it's divisible by 10.
And therefore those two
numbers are congruent.
It's very easy to tell when
two numbers are congruent
mod 10 because they just
have the same lower digit.
OK.
Another way to understand
congruency and what it's really
all about is the
so-called remainder lemma,
which sets that a is
congruent to b mod n, if
and only if a and b have the
same remainder on division
by n.
So let's work with
that definition.
We can conclude using this
formulation, equivalent
formulation, that 30
is equivalent to 12
mod 9 because the remainder
of 30 divided by 9,
well it's 3 times 9
is 27, remainder 3.
And the remainder
of 12 by 9 is 3.
So they do indeed have
the same remainder 3.
And they're congruent.
By the way, this equivalent sign
with the three horizontal bars
is read as both
equivalent and congruent.
And I will be bouncing back
between the two pronunciations
indiscriminately.
They are synonyms.
OK, let's think about proving
this remainder lemma just
for practice.
And in order to
fit on the slide,
I'm going to have to abbreviate
this idea of the remainder of b
divided by n with a
shorter notation r sub b n.
Just to fit.
OK.
So the if direction of
proving the remainder
limit that they're congruent
if and only if they
have the same remainder.
The if direction here
in an if and only if
is from right to left.
I've got to prove that if they
have the same remainder, then
they're congruent.
So there are the two
numbers, a and b.
By the division theorem,
or division algorithm,
they can each be expressed as
a quotient of a divided by n
times the quotient sub a plus
the remainder of a divided
by n.
And likewise, b can
be expressed in terms
of quotient and remainder.
And what we're given here is
that the remainders are equal.
But if the remainders
are equal, then clearly
when I subtract a minus b,
I get qa minus qb times n.
Sure enough, a minus
b is a multiple of n.
And that takes care of that one.
The only if direction now
goes from left to right.
So in the converse,
I'm going to assume
that n divides a minus b,
where a and b are expressed
in this form by the division
algorithm or division theorem.
So if n divides a minus
b, looking at a minus b
in that form what we're
seeing is that n divides
this qa minus qb times
n, plus the difference
of the remainders.
That's what I get just
by subtracting a and b.
But if you look at this
n divides that term,
the quotient times n.
And it therefore has to
divide the other term as well.
Because the only way
that n can divide
a sum, when it divides
one of the sum ands,
is if it divides
the other sum and.
So n divides ra minus the
remainder of 8 divided by n
from b divided by n.
But remember, these
are remainders.
So that means that they're
both in the interval from 0
to n minus 1 inclusive.
And the distance between them
has got to be less than 1.
So if n divides a number
that's between 0 and n minus 1,
that number has to be 0.
Because it's the only number
that n divides in there.
So in fact, the difference
of the remainders is 0.
And therefore, the
remainders are equal.
And we've knocked that one off.
So there it is restated.
The remainder lemma says
that they're congruent if
and only if they have
the same remainders.
And that's worth
putting a box around
to highlight this
crucial fact, which
could equally well have used as
the definition of congruence.
And then you'd prove
the division definition
that we began with.
Now some immediate consequences
of this remainder lemma
are that a congruence inherits
a lot of properties of equality.
Because it means
nothing more than that
the remainders are equal.
So for example, we can say
the congruence is symmetric,
meaning that if a is congruent
to b, then b is congruent to a.
And that's obvious
cause a congruent to b
means that a and b have
the same remainder.
So b and a have
the same remainder.
One that would actually
take a little bit of work
to prove from the
division definition--
not a lot, but a
little bit-- would
be that if a is congruent to b,
and b is congruent to c, then a
is congruent to c.
But we can read it is saying
the first says that a and b have
the same remainder.
The second says that b and
c have the same remainder.
So obviously a and c
have the same remainder.
And we've prove this
property that's known
as transitivity of congruence.
Another simple consequence
of the remainder theorem
is a little technical
result that's
enormously useful called
remainder lemma, which
says simply that a number is
congruent to its own remainder,
modulo n.
The proof is easy.
Let's prove it by showing
that a and the remainder of a
have the same remainder.
Well, what if I take
remainders of both sides,
the left hand side becomes the
remainder of a divided by n.
The right hand side is the
remainder of the remainder.
But the point is
that the remainder
is in the interval from 0 to n.
And that means when you take its
remainder mod and its itself.
And therefore the left
hand side is the remainder
of a divided by n, and
the right hand side
is also the remainder
of the a divided by n.
And we have proved
this corollary
that's the basis of
remainder arithmetic.
Which will basically
allow us whenever
we feel like it to replace
numbers by their remainders,
and that way keep
the numbers small.
And that also
merits a highlight.
OK.
Now, in addition to these
properties like equality
that congruence has, it
also interacts very well
the operations.
Which is why it's
called a congruence.
A congruence is an
equality-like relation
that respects the
operations that
are relevant to the discussion.
In this case, we're going to be
talking about plus and times.
And the first fact
about congruent
says that if a and
b are congruent,
then a plus c and b
plus c are congruent.
The proof of that follows
trivially from the definition.
Because the a
congruent to b mod,
n says that n divides a minus b.
And if n divides a minus b,
obviously n divides a plus
c minus b plus c.
Because a plus c minus b
plus c is equal to a minus b.
That one is deceptively trivial.
It's also the case that if a
is congruent to b, then a times
c is congruent to b times c.
This one takes a one line proof.
We're given that n
divides a minus b.
That certainly
implies that n divides
any multiple of a minus b.
So multiply it by c and
then apply distributivity,
and you discover
that n divides ac
minus bc, which means ac is
congruent to bc modulo n.
It's a small step
that I'm going to omit
to go from adding
the same constant
to both sides to adding
any two congruent
numbers to the same sides.
So if a is congruent to b
and c is congruent to d, then
in fact, a plus c is
congruent to b plus d.
So again, congruence is acting
a lot like ordinary equality.
If you add equals to
equals, you get equals.
And of course the same fact
applies to multiplication.
If you multiply equals by
equals, you get equals.
A corollary of this is that
if I have two numbers that
are congruent modulo n, then if
I have any kind of arithmetic
formula involving plus
and times and minus--
and what I want to know is what
it's equivalent to modulo n--
I can figure that out
freely substituting
a by a prime or a prime by a.
I can replace any number by a
number that it's congruent to,
and the final congruence
result of the formula
is going to remain unchanged.
So overall what this shows is
that arithmetic modulo n is
a lot like ordinary arithmetic.
And the other
crucial point thought
that follows from this
fact about remainders
is that because a is congruent
to the remainder of a divided
by n, then when I'm doing
arithmetic on congruences,
I can always keep the numbers
involved in the remainder
interval.
That is, in the remainder
range from 0 to n minus 1.
And we use this standard closed
open interval notation to mean
the interval from 0 to n.
So it's sometimes
used in analysis
to mean the real
interval of reals.
But we're always
talking about integers.
So this means-- the
integers that square bracket
means 0 is included.
And a round parenthesis
means that n is excluded.
So that's exactly a description
of the integers that
are greater and equal
to 0 and less than n.
Let's do an application of
this remainder arithmetic idea.
Suppose I want to figure out
what's 287 to the ninth power
modulo 4?
Well, for a start but if I
take the remainder of 287
divided by 4, it's not very
hard to check that that's 3.
And that means that 287 to the
ninth is congruent mod 4 to 3
to the ninth.
So already I got rid of
the three digit number,
the base of the
exponent, and replaced it
just by a one digit number, 3.
That's progress.
Well, we can make more
progress because 3 to the ninth
can be expressed as 3 squared,
squared, squared times 3,
right?
Because when you
iterate taking powers,
it means that the
exponents multiply.
So this is 3 to the 2 times
2 times 2, or 8, times 3--
which adds 1 to the
exponent-- or 9.
So that's simple
exponent arithmetic.
But notice that 3 squared is 9.
And 9 is congruent to 1 mod 4.
So that means I can
replace 3 squared by 1,
and the outer 2 squared stays.
It becomes 1 squared squared,
but that's 1 times 3.
And the punchline is that 287 to
the ninth is congruent to 3 mod
4 by a really easy calculation
that did not involve taking
anything to the ninth power.
