1
00:00:00,500 --> 00:00:03,380
PROFESSOR: Another way to talk
about congruence and remainder

2
00:00:03,380 --> 00:00:05,644
arithmetic is to work
strictly with remainders,

3
00:00:05,644 --> 00:00:08,060
which makes things a little
simpler because you don't have

4
00:00:08,060 --> 00:00:10,476
to worry about the fact that
the product of two remainders

5
00:00:10,476 --> 00:00:12,832
may, for example, be too
big to be a remainder.

6
00:00:12,832 --> 00:00:14,290
To knock it back
in range, you have

7
00:00:14,290 --> 00:00:15,790
to take the remainder again.

8
00:00:15,790 --> 00:00:18,000
And that's what
this abstract idea

9
00:00:18,000 --> 00:00:22,240
of the ring of integers
modulo n, the ring Z sub n,

10
00:00:22,240 --> 00:00:25,280
captures in a quite elegant way.

11
00:00:25,280 --> 00:00:27,660
So it's going to allow us to
talk strictly about equality

12
00:00:27,660 --> 00:00:28,980
instead of congruence.

13
00:00:28,980 --> 00:00:34,490
And let's remind ourselves that
the basic idea behind working

14
00:00:34,490 --> 00:00:38,760
with a remainder arithmetic
was that every time we

15
00:00:38,760 --> 00:00:41,210
got a number that was too
big to be a remainder,

16
00:00:41,210 --> 00:00:43,430
we just hit it with
the remainder operation

17
00:00:43,430 --> 00:00:45,530
again to bring it back in range.

18
00:00:45,530 --> 00:00:50,010
And so the operations in
Zn work exactly that way.

19
00:00:50,010 --> 00:00:52,820
The elements of Zn
are the remainders.

20
00:00:52,820 --> 00:00:56,890
That is, the numbers from
0, including 0, up to n,

21
00:00:56,890 --> 00:00:57,880
but not including n.

22
00:00:57,880 --> 00:01:01,760
So there are n of them from
0, 1, up through n minus 1.

23
00:01:01,760 --> 00:01:04,190
And the definitions of
the operations in Zn

24
00:01:04,190 --> 00:01:05,489
are given right here.

25
00:01:05,489 --> 00:01:07,320
Addition just
means take this sum

26
00:01:07,320 --> 00:01:09,790
but then take the
remainder immediately,

27
00:01:09,790 --> 00:01:11,250
just in case it's too big.

28
00:01:11,250 --> 00:01:15,200
And likewise, the product in
Zn is simply multiply them

29
00:01:15,200 --> 00:01:16,710
and take the remainder.

30
00:01:16,710 --> 00:01:20,090
This isn't really a
very dramatic idea,

31
00:01:20,090 --> 00:01:23,642
but it turns out to pay off
in making some things just

32
00:01:23,642 --> 00:01:25,350
a little bit easier
to say, because we're

33
00:01:25,350 --> 00:01:29,080
talking about equality
instead of congruence.

34
00:01:29,080 --> 00:01:32,100
So this package together,
this mathematical structure

35
00:01:32,100 --> 00:01:34,750
consisting of the integers
in this interval--

36
00:01:34,750 --> 00:01:36,640
remember this notation,
square bracket

37
00:01:36,640 --> 00:01:40,480
means inclusive and round
parenthesis means exclusive.

38
00:01:40,480 --> 00:01:42,850
So this includes zero,
it doesn't include n.

39
00:01:42,850 --> 00:01:46,280
The integers in that
interval, under the operations

40
00:01:46,280 --> 00:01:51,070
of plus and times modulo
Zn, as defined here,

41
00:01:51,070 --> 00:01:54,170
is called the ring
of integers Zn.

42
00:01:54,170 --> 00:01:56,880
So it's got two
operations and a bunch

43
00:01:56,880 --> 00:01:59,400
of things that are operated on.

44
00:01:59,400 --> 00:02:02,200
Now I guess it's
worth highlighting.

45
00:02:02,200 --> 00:02:04,330
That's what Zn is,
the ring of integers.

46
00:02:04,330 --> 00:02:06,780
Mod n, or modulo n.

47
00:02:06,780 --> 00:02:11,330
Now, arithmetic in Zn is really
just arithmetic-- congruence

48
00:02:11,330 --> 00:02:15,110
arithmetic, except that
it's equality now instead

49
00:02:15,110 --> 00:02:17,330
of congruence.

50
00:02:17,330 --> 00:02:20,970
So we can say, for example,
in Z7 that 3 plus 6

51
00:02:20,970 --> 00:02:24,800
is literally equal to 2
because, well, 3 plus 6 is 9,

52
00:02:24,800 --> 00:02:27,550
the remainder on
division by 7 is 2,

53
00:02:27,550 --> 00:02:30,570
and we go directly
to the two in Zn,

54
00:02:30,570 --> 00:02:33,651
suppressing the mention of
taking remainders and not even

55
00:02:33,651 --> 00:02:35,650
really having to think
about it, which is what's

56
00:02:35,650 --> 00:02:37,880
helpful about working with Zn.

57
00:02:37,880 --> 00:02:41,860
Likewise, 9 times 8 is
literally equal to 6 in Z11.

58
00:02:45,590 --> 00:02:48,360
So what's the connection between
the set of all the integers

59
00:02:48,360 --> 00:02:50,700
and the integers mod n?

60
00:02:50,700 --> 00:02:54,470
And we can state this
abstractly in the following way.

61
00:02:54,470 --> 00:02:57,980
Let's just, for convenience,
abbreviate the remainder of k

62
00:02:57,980 --> 00:03:00,160
on division by n as R of k.

63
00:03:00,160 --> 00:03:01,550
So n is fixed.

64
00:03:01,550 --> 00:03:03,990
And what's the connection
between Z and Zn?

65
00:03:03,990 --> 00:03:05,550
Well, it's fairly simple.

66
00:03:05,550 --> 00:03:08,420
If you take the
remainder of i plus j,

67
00:03:08,420 --> 00:03:11,970
that's literally equal
to taking the sum

68
00:03:11,970 --> 00:03:14,356
of the remainders in Zn.

69
00:03:14,356 --> 00:03:15,730
Once you've taken
the remainders,

70
00:03:15,730 --> 00:03:19,420
you're in the range of
numbers that Zn works with.

71
00:03:19,420 --> 00:03:22,030
And this sum, then, keeps
you in on the Zn side.

72
00:03:22,030 --> 00:03:24,760
Likewise, if you
take the remainder

73
00:03:24,760 --> 00:03:27,170
of a product of real
integers, that's

74
00:03:27,170 --> 00:03:31,140
literally equal to the product
of the remainders in Zn.

75
00:03:31,140 --> 00:03:32,940
This operation, by the
way, this connection

76
00:03:32,940 --> 00:03:35,020
between mathematical
structures, the structure

77
00:03:35,020 --> 00:03:37,220
of the integers
under plus and times

78
00:03:37,220 --> 00:03:40,640
and Zn under plus and times,
is called a homomorphism.

79
00:03:40,640 --> 00:03:44,570
R, in this case, is defining
a homomorphism from Z to Zn.

80
00:03:44,570 --> 00:03:46,432
That's a basic
concept in algebra

81
00:03:46,432 --> 00:03:47,890
that you'll learn
more about if you

82
00:03:47,890 --> 00:03:50,670
take some courses in algebra,
but I'm just mentioning it

83
00:03:50,670 --> 00:03:51,660
for cultural reasons.

84
00:03:51,660 --> 00:03:53,980
We're not going to
exploit it any further,

85
00:03:53,980 --> 00:03:55,970
or look further into this idea.

86
00:03:55,970 --> 00:03:57,446
OK.

87
00:03:57,446 --> 00:03:59,320
What's the connection
between equivalence mod

88
00:03:59,320 --> 00:04:01,684
n, or congruence mod n, and Zn?

89
00:04:01,684 --> 00:04:02,725
Well, it's fairly simple.

90
00:04:05,480 --> 00:04:08,660
In Zn, we convert
congruences into equalities.

91
00:04:08,660 --> 00:04:13,640
So i is congruent to j mod
n if and only if r of i

92
00:04:13,640 --> 00:04:16,880
is equal to r of j in Zn.

93
00:04:16,880 --> 00:04:19,140
And this is just a
rephrasing of the fact

94
00:04:19,140 --> 00:04:20,570
that two numbers
are congruent if

95
00:04:20,570 --> 00:04:24,080
and only if they have
the same remainder.

96
00:04:24,080 --> 00:04:27,250
Now once you've got this
self-contained system Zn,

97
00:04:27,250 --> 00:04:29,540
you can start talking
about algebraic rules

98
00:04:29,540 --> 00:04:30,710
that it satisfies.

99
00:04:30,710 --> 00:04:33,860
And now, they hold with equality
and they're pretty familiar.

100
00:04:33,860 --> 00:04:36,170
So let's look at some of
the rules for addition,

101
00:04:36,170 --> 00:04:38,730
for example, that
hold true in Zn.

102
00:04:38,730 --> 00:04:40,540
First of all, addition
is associative.

103
00:04:40,540 --> 00:04:44,150
i plus j plus k is
i plus j plus k.

104
00:04:44,150 --> 00:04:46,350
We have an identity
element, literally zero.

105
00:04:46,350 --> 00:04:49,100
Zero plus any i is i.

106
00:04:49,100 --> 00:04:53,100
We have a minus operation, an
inverse operation, with respect

107
00:04:53,100 --> 00:05:03,020
to addition, which is that--
how do I get back some slides?

108
00:05:03,020 --> 00:05:03,520
Excuse me.

109
00:05:03,520 --> 00:05:04,580
OK, let's keep going.

110
00:05:04,580 --> 00:05:08,830
I have an inverse operation,
which is that for every i,

111
00:05:08,830 --> 00:05:10,460
there's an element
called minus i.

112
00:05:10,460 --> 00:05:14,200
It's additive inverse such
that if you add i and minus i,

113
00:05:14,200 --> 00:05:15,680
you get zero.

114
00:05:15,680 --> 00:05:17,310
And finally,
commutativity, which

115
00:05:17,310 --> 00:05:20,100
is that i plus j is
the same as j plus i.

116
00:05:20,100 --> 00:05:22,110
You don't really need
to memorize these names,

117
00:05:22,110 --> 00:05:24,940
but you will probably
hear them a lot

118
00:05:24,940 --> 00:05:28,060
in various other contexts, and
especially in algebra courses,

119
00:05:28,060 --> 00:05:29,800
but even in terms of arithmetic.

120
00:05:29,800 --> 00:05:32,720
These are some of the basic
rules that addition satisfies.

121
00:05:32,720 --> 00:05:34,306
And in fact,
multiplication satisfies

122
00:05:34,306 --> 00:05:35,430
pretty much the same rules.

123
00:05:35,430 --> 00:05:38,340
Multiplication is
likewise associative.

124
00:05:38,340 --> 00:05:40,840
There's an identity for
multiplication called 1.

125
00:05:40,840 --> 00:05:43,110
1 times i is i.

126
00:05:43,110 --> 00:05:44,900
Multiplication is
also commutative.

127
00:05:44,900 --> 00:05:48,290
The one obvious omission
here is inverses.

128
00:05:48,290 --> 00:05:52,480
You can't count on there
being inverses in Zn.

129
00:05:52,480 --> 00:05:54,650
And finally,
there's an operation

130
00:05:54,650 --> 00:05:57,740
that connects addition
and multiplication called

131
00:05:57,740 --> 00:05:58,710
distributivity.

132
00:05:58,710 --> 00:06:02,890
Namely, i times j
plus k is ij plus ik,

133
00:06:02,890 --> 00:06:05,450
as you well know from
ordinary arithmetic.

134
00:06:05,450 --> 00:06:11,086
And this rule works fine for
remainders and working in Zn.

135
00:06:11,086 --> 00:06:12,960
As I said, the one thing
we have to watch out

136
00:06:12,960 --> 00:06:14,543
for, it shouldn't
be a surprise, is we

137
00:06:14,543 --> 00:06:17,570
know that you can't cancel with
respect to congruence mod n.

138
00:06:17,570 --> 00:06:20,540
And that's reflected in the fact
that you can't cancel in Zn.

139
00:06:20,540 --> 00:06:26,740
Namely, in Z12, for example, 3
times 2 is equal to 2 times 8.

140
00:06:26,740 --> 00:06:28,900
Again, 3 times 2 is
6, 2 times 8 is 16,

141
00:06:28,900 --> 00:06:31,570
you immediately take the
remainder to get back to 6.

142
00:06:31,570 --> 00:06:34,260
In Z12, these two
things are equal.

143
00:06:34,260 --> 00:06:37,960
But if you tried to cancel the
2, you'd conclude that 3 was 8,

144
00:06:37,960 --> 00:06:40,870
and neither 3-- 3 and
8 are different numbers

145
00:06:40,870 --> 00:06:45,110
in the range from 0 to 12,
and they're different in Z12.

146
00:06:45,110 --> 00:06:47,210
So you can't cancel 2.

147
00:06:47,210 --> 00:06:48,190
OK.

148
00:06:48,190 --> 00:06:50,850
Now the rules that we
already figured out

149
00:06:50,850 --> 00:06:52,640
for when you can
cancel in congruence

150
00:06:52,640 --> 00:06:56,090
translate directly over to
when you can cancel in Zn.

151
00:06:56,090 --> 00:06:58,610
And now there's a
standard abbreviation

152
00:06:58,610 --> 00:07:00,000
that's useful to use here.

153
00:07:00,000 --> 00:07:04,310
If I write Zn*, what I mean
is the elements in Zn that are

154
00:07:04,310 --> 00:07:06,420
relatively prime to n.

155
00:07:06,420 --> 00:07:09,320
The elements whose
GCD with n is 1.

156
00:07:09,320 --> 00:07:16,670
So what we have is the following
equivalent formulations of Zn*,

157
00:07:16,670 --> 00:07:19,300
which correspond to the facts
we've already figured out about

158
00:07:19,300 --> 00:07:19,800
congruence.

159
00:07:19,800 --> 00:07:25,280
Namely, an integer i in the
range from 0 to n is in Zn*

160
00:07:25,280 --> 00:07:28,300
if and only if the
GCD of i and n is 1,

161
00:07:28,300 --> 00:07:32,780
or i is cancelable in Zn,
or i has an inverse in Zn.

162
00:07:32,780 --> 00:07:34,720
All of these three
things are equivalent.

163
00:07:34,720 --> 00:07:40,260
They give you the sense that
Zn* is a kind of robust subset

164
00:07:40,260 --> 00:07:43,830
of Zn that you'd want
to be thinking about.

165
00:07:43,830 --> 00:07:46,730
And in fact, it's very valuable
to be paying attention to.

166
00:07:46,730 --> 00:07:48,870
What else do we know about Zn*?

167
00:07:48,870 --> 00:07:52,070
Well, the definition of
phi of n was the number

168
00:07:52,070 --> 00:07:54,720
of integers in the
interval from 0

169
00:07:54,720 --> 00:07:56,940
to n that are
relatively prime to n.

170
00:07:56,940 --> 00:08:00,870
Of course, that's
exactly the size of Zn*.

171
00:08:00,870 --> 00:08:04,980
So phi of n is simply the size
of that collection of elements.

172
00:08:04,980 --> 00:08:05,680
Not surprising.

173
00:08:05,680 --> 00:08:07,400
They were defined that way.

174
00:08:07,400 --> 00:08:09,152
So now I can restate
Euler's Theorem

175
00:08:09,152 --> 00:08:10,360
in a slightly convenient way.

176
00:08:10,360 --> 00:08:11,600
Instead of mentioning
congruence,

177
00:08:11,600 --> 00:08:12,933
we can just talk about equality.

178
00:08:12,933 --> 00:08:16,990
Euler's Theorem says that if you
raise a number k to the power

179
00:08:16,990 --> 00:08:20,570
phi of n, it's literally
equal to 1 in Zn,

180
00:08:20,570 --> 00:08:23,980
at least for those k's that
are relatively prime to n.

181
00:08:23,980 --> 00:08:27,600
That is, those k's
that are in Zn*.

182
00:08:27,600 --> 00:08:30,630
And it's going to turn out that
the proof of Euler's Theorem

183
00:08:30,630 --> 00:08:31,630
is actually pretty easy.

184
00:08:31,630 --> 00:08:33,129
It just follows in
a couple of steps

185
00:08:33,129 --> 00:08:34,700
from a couple of
simple observations.

186
00:08:34,700 --> 00:08:36,159
So let's start on those.

187
00:08:36,159 --> 00:08:40,270
So the first remark is that if I
have any subset, S, of elements

188
00:08:40,270 --> 00:08:43,720
in Zn-- I don't care whether
they are relatively prime to n

189
00:08:43,720 --> 00:08:47,180
or not-- if I multiply
each of them by k,

190
00:08:47,180 --> 00:08:50,110
this notation for
k times S means

191
00:08:50,110 --> 00:08:54,290
that I'm taking the set of
elements that are of the form

192
00:08:54,290 --> 00:08:57,750
k times an element of S
over all the elements of S.

193
00:08:57,750 --> 00:09:01,000
So kS, which is this
set of multiples

194
00:09:01,000 --> 00:09:04,930
of k-- multiples of
elements of S by k,

195
00:09:04,930 --> 00:09:07,330
has exactly the same size as S.

196
00:09:07,330 --> 00:09:08,120
Now, why is that?

197
00:09:08,120 --> 00:09:11,880
Well, this of course is only
true for k that are cancelable.

198
00:09:11,880 --> 00:09:15,895
But the Lemma is, no matter
what subset you take of Zn,

199
00:09:15,895 --> 00:09:19,060
if you multiplied every
one of them by an element

200
00:09:19,060 --> 00:09:23,550
that's cancelable in Zn*, you
get a set of the same size.

201
00:09:23,550 --> 00:09:28,470
And that's clear because how
could ks1 and ks2 be equal?

202
00:09:28,470 --> 00:09:30,526
Well, only if s1
and s2 were equal.

203
00:09:30,526 --> 00:09:32,150
Or another way to
say it is that if you

204
00:09:32,150 --> 00:09:35,730
had different elements
in S, s1 not equal to s2,

205
00:09:35,730 --> 00:09:37,870
when you multiply
them by k, you have

206
00:09:37,870 --> 00:09:42,780
to get different elements of
ks, because k is cancelable.

207
00:09:42,780 --> 00:09:43,280
OK.

208
00:09:43,280 --> 00:09:44,410
So that's an easy remark.

209
00:09:44,410 --> 00:09:46,030
Holds in general.

210
00:09:46,030 --> 00:09:50,690
Multiply any subset by
a cancelable element,

211
00:09:50,690 --> 00:09:53,960
and you get a new set
that's the same size.

212
00:09:53,960 --> 00:09:57,400
The second remark is that if
you look at numbers i and j that

213
00:09:57,400 --> 00:10:01,340
are in the interval
from 0 to n in Zn,

214
00:10:01,340 --> 00:10:06,660
then if you multiply
the two of them,

215
00:10:06,660 --> 00:10:11,650
then you're going to get an
element in Zn* if and only

216
00:10:11,650 --> 00:10:14,052
if the original two
elements were in Zn*.

217
00:10:14,052 --> 00:10:16,510
Well, let's just look at it in
the left to right direction,

218
00:10:16,510 --> 00:10:17,980
which is the only one we need.

219
00:10:17,980 --> 00:10:22,830
If i and j are
relatively prime to Zn*,

220
00:10:22,830 --> 00:10:26,910
then so is their product,
because if neither i nor j has

221
00:10:26,910 --> 00:10:30,480
a prime factor in common with
n, then their product obviously

222
00:10:30,480 --> 00:10:34,130
doesn't have a factor
in common with n.

223
00:10:34,130 --> 00:10:35,600
And then when you
take remainders,

224
00:10:35,600 --> 00:10:39,330
it's still going to be a
number whose GCD is the same.

225
00:10:39,330 --> 00:10:41,960
And so we have this
remark that if you

226
00:10:41,960 --> 00:10:44,180
multiply two
cancelable elements,

227
00:10:44,180 --> 00:10:45,490
you get a cancelable element.

228
00:10:45,490 --> 00:10:48,290
If you multiply two elements
relatively prime to Zn*,

229
00:10:48,290 --> 00:10:49,620
you get an element of Zn*.

230
00:10:49,620 --> 00:10:52,640
There's about-- every one
of these formulations of Zn*

231
00:10:52,640 --> 00:10:56,290
in terms of GCDs are
cancelable or inverse,

232
00:10:56,290 --> 00:10:58,800
and each of them gives a
separate and straightforward

233
00:10:58,800 --> 00:11:01,720
proof of the fact that
if i and j are in Zn*,

234
00:11:01,720 --> 00:11:02,937
then so is their product.

235
00:11:02,937 --> 00:11:04,520
Now it's worth
mentioning, by the way,

236
00:11:04,520 --> 00:11:06,590
that, in general,
their sum is not.

237
00:11:06,590 --> 00:11:12,160
If you add two elements that
are relatively prime to Zn*,

238
00:11:12,160 --> 00:11:14,910
even if their sum is non-zero,
you will typically get

239
00:11:14,910 --> 00:11:21,050
an element that is no longer
relatively prime to n.

240
00:11:21,050 --> 00:11:22,910
But for multiplication,
it works great,

241
00:11:22,910 --> 00:11:25,580
and that's what matters to us.

242
00:11:25,580 --> 00:11:26,080
OK.

243
00:11:26,080 --> 00:11:29,470
So as a corollary of this is
that I can actually conclude

244
00:11:29,470 --> 00:11:32,340
that, if I choose an
element that's cancelable,

245
00:11:32,340 --> 00:11:36,110
an element in Zn*, if I
take the whole set Zn*,

246
00:11:36,110 --> 00:11:38,780
all those elements that
are relatively prime to n,

247
00:11:38,780 --> 00:11:43,460
and I take multiples of k by
each of them, then, in fact,

248
00:11:43,460 --> 00:11:46,260
I get the same set, Zn*.

249
00:11:46,260 --> 00:11:52,140
And the proof of that is
really straightforward.

250
00:11:52,140 --> 00:11:53,716
Let's think about
it for a minute.

251
00:11:53,716 --> 00:11:55,590
Because what do I know
is that these two sets

252
00:11:55,590 --> 00:11:57,050
are the same size.

253
00:11:57,050 --> 00:12:00,060
kZn* and Zn* are the same size.

254
00:12:00,060 --> 00:12:02,890
As long as k is cancelable, I
don't even care that this was

255
00:12:02,890 --> 00:12:05,030
Zn*.

256
00:12:05,030 --> 00:12:08,810
On the other hand,
if k is in Zn*,

257
00:12:08,810 --> 00:12:12,940
k times Zn* only gives
you elements in Zn*.

258
00:12:12,940 --> 00:12:17,180
So kZn* is a subset
of the left-hand side,

259
00:12:17,180 --> 00:12:20,220
and it's the same size by
the Lemma that says that

260
00:12:20,220 --> 00:12:22,900
multiplying by k
preserves sizes.

261
00:12:22,900 --> 00:12:24,540
So they have to be equal.

262
00:12:24,540 --> 00:12:27,670
So basically what that means
is that if you take all

263
00:12:27,670 --> 00:12:31,230
the elements in Z*, all the
elements relatively prime to n,

264
00:12:31,230 --> 00:12:34,600
and you take another one of
them, pick one out of that set,

265
00:12:34,600 --> 00:12:39,280
and multiply every element
in the set by that element k,

266
00:12:39,280 --> 00:12:42,170
if you had them lined up
in one order beforehand,

267
00:12:42,170 --> 00:12:45,220
when you multiplied by k you
get exactly the same elements

268
00:12:45,220 --> 00:12:47,100
but just reordered.

269
00:12:47,100 --> 00:12:50,620
That is, multiplying by k
has the effect of permuting

270
00:12:50,620 --> 00:12:54,410
the elements of Zn*.

271
00:12:54,410 --> 00:12:55,750
Let's look at an example.

272
00:12:55,750 --> 00:12:57,700
So let's look at Z9.

273
00:12:57,700 --> 00:13:00,900
And we know that phi of 9,
by the previous formula,

274
00:13:00,900 --> 00:13:02,490
is 3 squared minus 3, or 6.

275
00:13:02,490 --> 00:13:05,840
There are going to be 6 elements
from 0 to n that are relatively

276
00:13:05,840 --> 00:13:10,060
prime to 9, and
that comprise Zn*.

277
00:13:10,060 --> 00:13:11,740
So let's look at what they are.

278
00:13:11,740 --> 00:13:13,580
So you can do-- check
the calculation.

279
00:13:13,580 --> 00:13:18,447
But Zn* is exactly the
elements 1, 2, 4, 5, 7, 8.

280
00:13:18,447 --> 00:13:20,280
We know we got them all
because there's only

281
00:13:20,280 --> 00:13:21,696
supposed to be six
of them, and we

282
00:13:21,696 --> 00:13:24,460
can check that those are
all relatively prime to 9.

283
00:13:24,460 --> 00:13:26,720
None of them has 3 as a divisor.

284
00:13:26,720 --> 00:13:29,840
Now what happens, for example,
if I multiply them all by 2?

285
00:13:29,840 --> 00:13:32,210
Two is another good
number-- it's right here--

286
00:13:32,210 --> 00:13:33,790
that's in Zn*.

287
00:13:33,790 --> 00:13:35,850
And multiplying them by
2, well, let's check.

288
00:13:35,850 --> 00:13:39,750
2 times 1 is 2, 2 times
2 is 4, 2 times 4 is 8,

289
00:13:39,750 --> 00:13:45,060
2 times 5 is 1-- because it's 10
with a remainder of 1-- 2 times

290
00:13:45,060 --> 00:13:47,940
7 is 14-- translates
into 5-- 2 times 8

291
00:13:47,940 --> 00:13:51,140
is 16-- [INAUDIBLE]
translates into 7.

292
00:13:51,140 --> 00:13:52,720
And, as claimed, look at this.

293
00:13:52,720 --> 00:13:55,350
Here's 2, 4, 8, 1, 5, 7.

294
00:13:55,350 --> 00:13:58,580
It's the same numbers
as 1, 2, 4, 5, 7, 8,

295
00:13:58,580 --> 00:14:00,070
just in a different order.

296
00:14:00,070 --> 00:14:02,040
Let's do one more example.

297
00:14:02,040 --> 00:14:03,840
Let's try multiplying by 7.

298
00:14:03,840 --> 00:14:06,130
That's another respectable
element over here.

299
00:14:06,130 --> 00:14:14,331
7 times 1 is 7, 7 times 2 is
14, which means it's 5 in Z9.

300
00:14:14,331 --> 00:14:15,850
4 times 7 is 28.

301
00:14:15,850 --> 00:14:20,490
Well, 3 times 7 is 27, so
that leaves a remainder of 1.

302
00:14:20,490 --> 00:14:22,590
And 4 times 7 is 1 in Z9.

303
00:14:22,590 --> 00:14:27,050
Likewise, 5 times 7 is 8, 7
times 7 is 4, and 7 times 8

304
00:14:27,050 --> 00:14:30,240
is 56, which translates to 2.

305
00:14:30,240 --> 00:14:34,130
And sure enough, as claimed, I
see the same numbers, 7, 5, 1,

306
00:14:34,130 --> 00:14:38,390
8, 4, 2, just these
numbers scrambled in order.

307
00:14:38,390 --> 00:14:43,220
They're permuted, which is the
outcome of multiplying by 7.

308
00:14:43,220 --> 00:14:44,640
OK.

309
00:14:44,640 --> 00:14:46,410
So let's go back.

310
00:14:46,410 --> 00:14:48,930
What we've just illustrated is
this fact that we've already

311
00:14:48,930 --> 00:14:53,900
concluded that, if you take
Zn* and you multiply it

312
00:14:53,900 --> 00:14:57,320
by an element k in Zn*, you
get the same set in a different

313
00:14:57,320 --> 00:14:57,820
order.

314
00:14:57,820 --> 00:15:02,450
So Zn* is equal to k times Zn*.

315
00:15:02,450 --> 00:15:05,920
And now we're on the brink
of proving Euler's Theorem.

316
00:15:05,920 --> 00:15:07,890
Because what I want
to do is say, look,

317
00:15:07,890 --> 00:15:09,430
these two sets are the same.

318
00:15:09,430 --> 00:15:12,390
Let's multiply all the
elements on the left

319
00:15:12,390 --> 00:15:15,290
together, and multiply all the
elements on the right together.

320
00:15:15,290 --> 00:15:17,780
Let's take the product
of those elements.

321
00:15:17,780 --> 00:15:22,940
So let's take the product of Zn*
and compare it to the product

322
00:15:22,940 --> 00:15:24,260
of kZn*.

323
00:15:24,260 --> 00:15:26,870
So big pi here is
indicating the product

324
00:15:26,870 --> 00:15:29,580
of all of the elements in
this set, the product of all

325
00:15:29,580 --> 00:15:31,320
of the elements in this set.

326
00:15:31,320 --> 00:15:35,420
Well, let's look at
the set on the right.

327
00:15:35,420 --> 00:15:39,950
This is the product of k
times all the elements in Z*.

328
00:15:39,950 --> 00:15:41,610
Well how many
elements are there?

329
00:15:41,610 --> 00:15:44,510
Phi of n elements in
Z*, by definition.

330
00:15:44,510 --> 00:15:47,090
And let's factor
out all the k's.

331
00:15:47,090 --> 00:15:51,500
So this expression here, the
product of k times each element

332
00:15:51,500 --> 00:15:56,100
in Zn*, is the same as the
product of the elements in Zn*

333
00:15:56,100 --> 00:15:59,380
times k to as many
elements as there were,

334
00:15:59,380 --> 00:16:00,690
namely k to the phi of n.

335
00:16:00,690 --> 00:16:03,150
I'm just factoring k
out of this product.

336
00:16:03,150 --> 00:16:05,070
And there's my k
to the phi of n.

337
00:16:05,070 --> 00:16:06,840
And now look what I got here.

338
00:16:06,840 --> 00:16:10,250
That's pi Zn*,
and that's pi Zn*.

339
00:16:10,250 --> 00:16:13,620
What do I know about
multiplying elements in Zn*?

340
00:16:13,620 --> 00:16:15,330
They're in Zn*.

341
00:16:15,330 --> 00:16:18,440
This product will be some
other element is Zn*.

342
00:16:18,440 --> 00:16:19,750
So will this product.

343
00:16:19,750 --> 00:16:21,720
But what do I know about Zn*?

344
00:16:21,720 --> 00:16:23,210
They're cancelable.

345
00:16:23,210 --> 00:16:26,520
So just looking-- ignoring
the middle term now,

346
00:16:26,520 --> 00:16:29,660
what I'm concluding is that the
product of Zn* is k to the phi

347
00:16:29,660 --> 00:16:31,440
of n times the product of Zn*.

348
00:16:31,440 --> 00:16:33,830
Let's cancel those
cancelable terms.

349
00:16:33,830 --> 00:16:34,730
And I'm done.

350
00:16:34,730 --> 00:16:36,500
I've just figured
out that 1, which

351
00:16:36,500 --> 00:16:39,520
is the result of canceling
the term on the left,

352
00:16:39,520 --> 00:16:41,440
is equal to k to the phi of n.

353
00:16:41,440 --> 00:16:45,850
And we have successfully
proved Euler's Theorem,

354
00:16:45,850 --> 00:16:49,270
which is what we were
aiming for in this segment.