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PROFESSOR: The
idea of congruence

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was introduced to
the world by Gauss

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in the early 18th century.

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You've heard of him
before, I think.

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He's responsible for some
work on magnetism also.

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And it turns out that this
idea, after several centuries,

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remains an active field of
application and research.

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And in particular,
in computer science

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it's used significantly
in crypto,

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which is what we're going to be
leading up to now in this unit.

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It's plays a role
in hashing, which

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is a key method for
managing data in memory.

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But we are not going to
go into that application.

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Anyway, the definition of
congruence is real simple.

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Congruence is a relation
between two numbers, a and b.

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It's determined by
another parameter

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n, where n is considered
to be greater than one.

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All of these, as
usual, are integers.

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And the definition
is simply that a

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is congruent to b mod n if n
divides a minus b or a minus

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b is a multiple of n.

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So that's a key
definition to remember.

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There's other ways to define it.

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We'll see very shortly
an equivalent formulation

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that could equally well have
been used as a definition.

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But this is a standard one.

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A is equivalent to b means that
a minus b is a multiple of n.

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Well let's just practice.

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30 is equivalent to 12 mod
9 because 30 minus 12 is 18,

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and 9 divides 18.

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OK.

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An immediate
application is that does

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this number with a lot
of 6's is ending in a 3

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is equivalent to
788253 modulo 10.

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Now why is that?

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Well, there's a
very simple reason.

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If you think about subtracting
the 6 number ending

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in 3 from the 7 number ending
in 3, what you can immediately

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see without doing much of
any of the subtraction--

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just do the low order digits--
when you subtract these,

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you're going to get a
number that ends in 0.

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Which means that
it's divisible by 10.

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And therefore those two
numbers are congruent.

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It's very easy to tell when
two numbers are congruent

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mod 10 because they just
have the same lower digit.

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OK.

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Another way to understand
congruency and what it's really

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all about is the
so-called remainder lemma,

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which sets that a is
congruent to b mod n, if

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and only if a and b have the
same remainder on division

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by n.

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So let's work with
that definition.

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We can conclude using this
formulation, equivalent

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formulation, that 30
is equivalent to 12

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mod 9 because the remainder
of 30 divided by 9,

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well it's 3 times 9
is 27, remainder 3.

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And the remainder
of 12 by 9 is 3.

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So they do indeed have
the same remainder 3.

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And they're congruent.

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By the way, this equivalent sign
with the three horizontal bars

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is read as both
equivalent and congruent.

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And I will be bouncing back
between the two pronunciations

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indiscriminately.

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They are synonyms.

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OK, let's think about proving
this remainder lemma just

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for practice.

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And in order to
fit on the slide,

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I'm going to have to abbreviate
this idea of the remainder of b

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divided by n with a
shorter notation r sub b n.

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Just to fit.

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OK.

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So the if direction of
proving the remainder

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limit that they're congruent
if and only if they

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have the same remainder.

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The if direction here
in an if and only if

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is from right to left.

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I've got to prove that if they
have the same remainder, then

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they're congruent.

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So there are the two
numbers, a and b.

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By the division theorem,
or division algorithm,

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they can each be expressed as
a quotient of a divided by n

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times the quotient sub a plus
the remainder of a divided

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by n.

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And likewise, b can
be expressed in terms

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of quotient and remainder.

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And what we're given here is
that the remainders are equal.

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But if the remainders
are equal, then clearly

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when I subtract a minus b,
I get qa minus qb times n.

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Sure enough, a minus
b is a multiple of n.

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And that takes care of that one.

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The only if direction now
goes from left to right.

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So in the converse,
I'm going to assume

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that n divides a minus b,
where a and b are expressed

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in this form by the division
algorithm or division theorem.

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So if n divides a minus
b, looking at a minus b

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in that form what we're
seeing is that n divides

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this qa minus qb times
n, plus the difference

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of the remainders.

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That's what I get just
by subtracting a and b.

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But if you look at this
n divides that term,

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the quotient times n.

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And it therefore has to
divide the other term as well.

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Because the only way
that n can divide

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a sum, when it divides
one of the sum ands,

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is if it divides
the other sum and.

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So n divides ra minus the
remainder of 8 divided by n

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from b divided by n.

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But remember, these
are remainders.

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So that means that they're
both in the interval from 0

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to n minus 1 inclusive.

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And the distance between them
has got to be less than 1.

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So if n divides a number
that's between 0 and n minus 1,

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that number has to be 0.

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Because it's the only number
that n divides in there.

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So in fact, the difference
of the remainders is 0.

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And therefore, the
remainders are equal.

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And we've knocked that one off.

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So there it is restated.

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The remainder lemma says
that they're congruent if

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and only if they have
the same remainders.

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And that's worth
putting a box around

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to highlight this
crucial fact, which

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could equally well have used as
the definition of congruence.

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And then you'd prove
the division definition

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that we began with.

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Now some immediate consequences
of this remainder lemma

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are that a congruence inherits
a lot of properties of equality.

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Because it means
nothing more than that

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the remainders are equal.

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So for example, we can say
the congruence is symmetric,

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meaning that if a is congruent
to b, then b is congruent to a.

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And that's obvious
cause a congruent to b

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means that a and b have
the same remainder.

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So b and a have
the same remainder.

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One that would actually
take a little bit of work

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to prove from the
division definition--

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not a lot, but a
little bit-- would

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be that if a is congruent to b,
and b is congruent to c, then a

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is congruent to c.

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But we can read it is saying
the first says that a and b have

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the same remainder.

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The second says that b and
c have the same remainder.

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So obviously a and c
have the same remainder.

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And we've prove this
property that's known

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as transitivity of congruence.

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Another simple consequence
of the remainder theorem

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is a little technical
result that's

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enormously useful called
remainder lemma, which

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says simply that a number is
congruent to its own remainder,

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modulo n.

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The proof is easy.

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Let's prove it by showing
that a and the remainder of a

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have the same remainder.

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Well, what if I take
remainders of both sides,

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the left hand side becomes the
remainder of a divided by n.

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The right hand side is the
remainder of the remainder.

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But the point is
that the remainder

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is in the interval from 0 to n.

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And that means when you take its
remainder mod and its itself.

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And therefore the left
hand side is the remainder

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of a divided by n, and
the right hand side

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is also the remainder
of the a divided by n.

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And we have proved
this corollary

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that's the basis of
remainder arithmetic.

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Which will basically
allow us whenever

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we feel like it to replace
numbers by their remainders,

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and that way keep
the numbers small.

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And that also
merits a highlight.

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OK.

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Now, in addition to these
properties like equality

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that congruence has, it
also interacts very well

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the operations.

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Which is why it's
called a congruence.

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A congruence is an
equality-like relation

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that respects the
operations that

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are relevant to the discussion.

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In this case, we're going to be
talking about plus and times.

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And the first fact
about congruent

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says that if a and
b are congruent,

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then a plus c and b
plus c are congruent.

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The proof of that follows
trivially from the definition.

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Because the a
congruent to b mod,

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n says that n divides a minus b.

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And if n divides a minus b,
obviously n divides a plus

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c minus b plus c.

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Because a plus c minus b
plus c is equal to a minus b.

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That one is deceptively trivial.

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It's also the case that if a
is congruent to b, then a times

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c is congruent to b times c.

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This one takes a one line proof.

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We're given that n
divides a minus b.

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That certainly
implies that n divides

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any multiple of a minus b.

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So multiply it by c and
then apply distributivity,

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and you discover
that n divides ac

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minus bc, which means ac is
congruent to bc modulo n.

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It's a small step
that I'm going to omit

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to go from adding
the same constant

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to both sides to adding
any two congruent

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numbers to the same sides.

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So if a is congruent to b
and c is congruent to d, then

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in fact, a plus c is
congruent to b plus d.

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So again, congruence is acting
a lot like ordinary equality.

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If you add equals to
equals, you get equals.

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And of course the same fact
applies to multiplication.

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If you multiply equals by
equals, you get equals.

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A corollary of this is that
if I have two numbers that

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are congruent modulo n, then if
I have any kind of arithmetic

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formula involving plus
and times and minus--

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and what I want to know is what
it's equivalent to modulo n--

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I can figure that out
freely substituting

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a by a prime or a prime by a.

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I can replace any number by a
number that it's congruent to,

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and the final congruence
result of the formula

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is going to remain unchanged.

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So overall what this shows is
that arithmetic modulo n is

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a lot like ordinary arithmetic.

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And the other
crucial point thought

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that follows from this
fact about remainders

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is that because a is congruent
to the remainder of a divided

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by n, then when I'm doing
arithmetic on congruences,

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I can always keep the numbers
involved in the remainder

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interval.

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That is, in the remainder
range from 0 to n minus 1.

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And we use this standard closed
open interval notation to mean

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the interval from 0 to n.

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So it's sometimes
used in analysis

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to mean the real
interval of reals.

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But we're always
talking about integers.

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So this means-- the
integers that square bracket

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means 0 is included.

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And a round parenthesis
means that n is excluded.

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So that's exactly a description
of the integers that

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are greater and equal
to 0 and less than n.

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Let's do an application of
this remainder arithmetic idea.

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Suppose I want to figure out
what's 287 to the ninth power

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modulo 4?

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Well, for a start but if I
take the remainder of 287

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divided by 4, it's not very
hard to check that that's 3.

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And that means that 287 to the
ninth is congruent mod 4 to 3

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to the ninth.

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So already I got rid of
the three digit number,

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the base of the
exponent, and replaced it

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just by a one digit number, 3.

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That's progress.

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00:12:15,310 --> 00:12:17,660
Well, we can make more
progress because 3 to the ninth

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00:12:17,660 --> 00:12:21,861
can be expressed as 3 squared,
squared, squared times 3,

248
00:12:21,861 --> 00:12:22,360
right?

249
00:12:22,360 --> 00:12:25,870
Because when you
iterate taking powers,

250
00:12:25,870 --> 00:12:27,490
it means that the
exponents multiply.

251
00:12:27,490 --> 00:12:31,280
So this is 3 to the 2 times
2 times 2, or 8, times 3--

252
00:12:31,280 --> 00:12:33,580
which adds 1 to the
exponent-- or 9.

253
00:12:33,580 --> 00:12:36,630
So that's simple
exponent arithmetic.

254
00:12:36,630 --> 00:12:43,310
But notice that 3 squared is 9.

255
00:12:43,310 --> 00:12:46,090
And 9 is congruent to 1 mod 4.

256
00:12:46,090 --> 00:12:50,400
So that means I can
replace 3 squared by 1,

257
00:12:50,400 --> 00:12:53,510
and the outer 2 squared stays.

258
00:12:53,510 --> 00:12:57,740
It becomes 1 squared squared,
but that's 1 times 3.

259
00:12:57,740 --> 00:13:03,300
And the punchline is that 287 to
the ninth is congruent to 3 mod

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00:13:03,300 --> 00:13:09,880
4 by a really easy calculation
that did not involve taking

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00:13:09,880 --> 00:13:12,460
anything to the ninth power.