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We're on problem number 7.

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If the average of x and 3x is
12, what is the value of x?

4
00:00:08,189 --> 00:00:12,370
So the average, so x plus 3x--
and I'm averaging two numbers

5
00:00:12,370 --> 00:00:15,270
so divide by 2-- that's going
to be equal to 12.

6
00:00:15,270 --> 00:00:16,180
So we just solve for this.

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00:00:16,180 --> 00:00:19,800
Multiply both sides of the
equation by 2 and you get 2

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00:00:19,800 --> 00:00:22,090
times, times 2, this
cancels with this.

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00:00:22,090 --> 00:00:26,020
You get x plus 3x
is equal to 24.

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00:00:26,020 --> 00:00:26,890
And what's x plus 3x.

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00:00:26,890 --> 00:00:28,670
That's 4x, right?

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00:00:28,670 --> 00:00:30,645
4x is equal to 24.

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x is equal to 6, and
that's choice C.

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Next problem.

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Problem 8.

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00:00:42,970 --> 00:00:46,700
At Maple Creek High School, some
members of the chess club

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are also on the swim team, and
no members of the swim team

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00:00:50,320 --> 00:00:52,130
are tenth graders.

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Which of the following
must be true.

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00:00:54,830 --> 00:00:57,220
This seems like it'll call
for a Venn diagram.

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00:00:57,220 --> 00:01:02,680
So let's say that that
represents the chess club.

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And they say some members
of the chess club

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00:01:04,790 --> 00:01:06,210
are on the swim team.

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00:01:06,210 --> 00:01:08,930
So some members are
on the swim team.

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00:01:08,930 --> 00:01:12,620
Maybe I should put the swim
team in like blue.

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00:01:12,620 --> 00:01:16,100
So let's say the swim team.

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That's the swim team.

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00:01:19,740 --> 00:01:21,510
And these are the members,
right, that are

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in both right here.

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But then it tells us no members
of the swim team are

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tenth graders.

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So if I draw another circle
for the tenth graders, it

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00:01:32,970 --> 00:01:35,560
can't intersect with the swim
team, but it could intersect

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00:01:35,560 --> 00:01:36,080
with the chess team.

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I don't know.

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I mean it could be like that.

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That could be tenth graders.

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It could be like that.

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Or it could be out
here some place.

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00:01:45,930 --> 00:01:47,150
But we don't know.

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00:01:47,150 --> 00:01:49,880
There could be chess and tenth
graders, just not the same

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00:01:49,880 --> 00:01:52,440
people who are on
the swim team.

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00:01:52,440 --> 00:01:53,690
So let's see.

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00:01:53,690 --> 00:01:55,820


47
00:01:55,820 --> 00:01:57,180
So which of the following
must be true?

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00:01:57,180 --> 00:01:58,960
No members of the chess club
are tenth graders.

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00:01:58,960 --> 00:01:59,370
No.

50
00:01:59,370 --> 00:02:01,800
This is a situation where you
could have some members of the

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00:02:01,800 --> 00:02:03,500
chess club who aren't
on the swim team who

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00:02:03,500 --> 00:02:05,420
could be tenth graders.

53
00:02:05,420 --> 00:02:09,039
B, some members of the chess
club are tenth graders.

54
00:02:09,039 --> 00:02:11,840
Well some members could be, but
we don't know for sure.

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This could be tenth grade.

56
00:02:12,880 --> 00:02:13,690
We don't know.

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This could be the tenth grade
kind of set or this could be

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the tenth grade.

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00:02:16,970 --> 00:02:19,590
There might be no tenth graders
in either the chess

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00:02:19,590 --> 00:02:20,310
team or the swim team.

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00:02:20,310 --> 00:02:21,890
We don't know for sure.

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And then choice C, some members
of the chess club are

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not tenth graders.

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This we know for sure.

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How do we know it for sure?

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Because these kids who are on
both, they're in the chess

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club, but they're also
on the swim team.

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00:02:39,210 --> 00:02:41,960
The fact that they're in swim
team, we know that they can't

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be tenth graders.

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So this is some members of the
chess club-- this little

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intersection here-- that
are not tenth graders.

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00:02:50,330 --> 00:02:52,440
So choice C is the
correct choice.

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00:02:52,440 --> 00:02:55,280


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00:02:55,280 --> 00:02:56,530
Next problem.

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If 3x plus n is equal
to x plus 1, what is

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00:03:08,380 --> 00:03:09,540
n in terms of x?

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00:03:09,540 --> 00:03:11,555
So we essentially just
solve for n.

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Let's subtract 3x
from both sides.

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00:03:13,790 --> 00:03:17,100
You get n is equal to--
what's x minus 3x?

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00:03:17,100 --> 00:03:18,580
It's minus 2x.

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And n plus 1.

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00:03:21,000 --> 00:03:21,710
And we're done.

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00:03:21,710 --> 00:03:23,870
And that choice isn't there, but
if you just switch these

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00:03:23,870 --> 00:03:27,000
two terms you just get that
equals 1 minus 2x and

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00:03:27,000 --> 00:03:28,900
that's choice D.

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00:03:28,900 --> 00:03:31,850
Pretty quick problem, especially
for one that's the

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00:03:31,850 --> 00:03:32,450
ninth problem.

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00:03:32,450 --> 00:03:34,110
They normally get a little
harder by this point.

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Problem 10.

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If k is a positive integer, let
k be defined as a set of

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00:03:41,290 --> 00:03:42,470
all multiples of k.

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So k with a square around it
is equal to the set of

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00:03:46,480 --> 00:03:52,490
multiples of k.

95
00:03:52,490 --> 00:03:56,900
All of the numbers in which of
the following sets are also in

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00:03:56,900 --> 00:03:59,770
all three of the set-- OK.

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00:03:59,770 --> 00:04:04,690
All of the numbers in which of
the following sets are also in

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00:04:04,690 --> 00:04:10,900
all three of the sets
of 2, 3 and 5?

99
00:04:10,900 --> 00:04:22,330
So the what they're saying is 2,
3, 5, this donates all the

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00:04:22,330 --> 00:04:23,580
multiples of 2.

101
00:04:23,580 --> 00:04:26,730


102
00:04:26,730 --> 00:04:30,450
This is all multiples of 3.

103
00:04:30,450 --> 00:04:37,340
This is all multiples of 5.

104
00:04:37,340 --> 00:04:41,800
So what they're essentially
saying is let's find a number

105
00:04:41,800 --> 00:04:45,970
where all of its multiples, all
of this number's multiples

106
00:04:45,970 --> 00:04:49,740
are also going to be multiples
of each of these.

107
00:04:49,740 --> 00:04:53,290
So it has to be a multiple--
so every number that--

108
00:04:53,290 --> 00:04:55,920
whatever this mystery number
is, let's call it x-- every

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00:04:55,920 --> 00:05:00,790
multiple of x has to be a
multiple of 2, 3 and 5.

110
00:05:00,790 --> 00:05:05,950
Well the simple way is if x is a
multiple of 2, 3 and 5, then

111
00:05:05,950 --> 00:05:07,340
every multiple of x
is going to be a

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00:05:07,340 --> 00:05:08,970
multiple of 2, 3 and 5.

113
00:05:08,970 --> 00:05:11,160
So what's 2 times 3 times 5?

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00:05:11,160 --> 00:05:13,680
It's 2 times 3 times 5.

115
00:05:13,680 --> 00:05:16,090
That's 6 times 5, that's 30.

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00:05:16,090 --> 00:05:20,160
So 30 is a multiple of all of
them, so any multiple of 30

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00:05:20,160 --> 00:05:21,990
will be a multiple
of all of these.

118
00:05:21,990 --> 00:05:25,050
When we look at the choices
we don't see 30.

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00:05:25,050 --> 00:05:27,360
But do we see any other
number that is a

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00:05:27,360 --> 00:05:30,010
multiple of 2, 3 and 5?

121
00:05:30,010 --> 00:05:32,160
Well sure, 60 is, right?

122
00:05:32,160 --> 00:05:33,620
We just multiply by 2 again.

123
00:05:33,620 --> 00:05:36,100
But 60 is still a multiple
of 2, 3 and 5.

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00:05:36,100 --> 00:05:38,530
If you were to do 2, 4, 6, 8 all
the way you'd get 60, if

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00:05:38,530 --> 00:05:41,260
you go 3, 9, 12, 15 all the
way, you'd get to 60.

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00:05:41,260 --> 00:05:44,580
You go 5, 10, 15, 20,
25, you'd get to 60.

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00:05:44,580 --> 00:05:46,630
So 60 is a multiple
of all of them.

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00:05:46,630 --> 00:05:50,600
So what we're saying is-- so
what's the set of all the

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00:05:50,600 --> 00:05:51,410
multiples of 60?

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00:05:51,410 --> 00:05:58,260
It's 60, 120, 180, 240,
et cetera, right?

131
00:05:58,260 --> 00:06:02,510
And all of these numbers are
in each of these sets.

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00:06:02,510 --> 00:06:06,020
Because all of these numbers are
multiples of 2, 3 and 5.

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00:06:06,020 --> 00:06:07,190
So our answer is 60.

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00:06:07,190 --> 00:06:09,320
If you look at the other
choices, some of them are

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divisible by 5, some are
divisible by 2 or 3,

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00:06:11,740 --> 00:06:13,100
some are 3 and 5.

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00:06:13,100 --> 00:06:17,870
But none of them are divisible
by 2, 3 and 5, only 60 is.

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00:06:17,870 --> 00:06:19,120
Next problem.

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00:06:21,760 --> 00:06:23,890
That problem was a little hard
to read initially though.

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That's how they confuse you.

142
00:06:25,140 --> 00:06:31,210


143
00:06:31,210 --> 00:06:33,383
So we're going to go from A to
D-- I should have drawn all

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00:06:33,383 --> 00:06:38,570
the lines first. Let me draw the
lines first. It's like a

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00:06:38,570 --> 00:06:40,741
hexagon kind of.

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00:06:40,741 --> 00:06:43,900
The top, the outside of
the hexagon there.

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00:06:47,100 --> 00:06:53,330
A, B, C, D, E, F.

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00:06:53,330 --> 00:06:54,740
And then this is the origin.

150
00:06:54,740 --> 00:06:59,640
And the figure above,
AD is equal to BE.

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00:06:59,640 --> 00:07:00,020
Oh, no, no.

152
00:07:00,020 --> 00:07:00,810
They don't tell us that.

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00:07:00,810 --> 00:07:01,820
I'm hallucinating.

154
00:07:01,820 --> 00:07:05,720
In the figure above AD, BE,
and CF intersect at 0.0.

155
00:07:05,720 --> 00:07:07,880
The intersect's here
at the origin.

156
00:07:07,880 --> 00:07:12,710
If the measure of AOB, the
measure of that, is 80

157
00:07:12,710 --> 00:07:22,520
degrees, and CF bisects
BOD, so it

158
00:07:22,520 --> 00:07:26,110
bisects this larger angle.

159
00:07:26,110 --> 00:07:28,880
CF bisect BOD, that angle.

160
00:07:28,880 --> 00:07:32,230
So that tells us that
this angle has to be

161
00:07:32,230 --> 00:07:34,100
equal to this angle.

162
00:07:34,100 --> 00:07:35,520
That's the definition of
bisecting an angle.

163
00:07:35,520 --> 00:07:37,310
You're splitting this larger
angle in half.

164
00:07:37,310 --> 00:07:41,110
So these angles have to be
equal to each other.

165
00:07:41,110 --> 00:07:43,270
So what is the measure of EOF?

166
00:07:43,270 --> 00:07:47,600


167
00:07:47,600 --> 00:07:51,770
So we want to figure
out this angle.

168
00:07:51,770 --> 00:07:54,285
Well this angle is opposite to
this angle, so they're going

169
00:07:54,285 --> 00:07:54,840
to be equal.

170
00:07:54,840 --> 00:07:56,970
So if we can figure out
this angle we're done.

171
00:07:56,970 --> 00:07:59,370
So let's call this angle x.

172
00:07:59,370 --> 00:08:03,770
If that angle's x this
angle is also x.

173
00:08:03,770 --> 00:08:05,900
This x, this x, and this
80 degrees, they're all

174
00:08:05,900 --> 00:08:10,440
supplementary because they all
go halfway around the circle.

175
00:08:10,440 --> 00:08:16,140
So x plus x plus 80 is going
to be equal to 180 degrees.

176
00:08:16,140 --> 00:08:20,090
2x plus 80 is equal to 180.

177
00:08:20,090 --> 00:08:24,590
2x is equal to 100,
x is equal to 50.

178
00:08:24,590 --> 00:08:29,390
And as we said before, x is
equal to 50, the angle EOF,

179
00:08:29,390 --> 00:08:31,750
which you're trying to figure
out, is opposite to it so it's

180
00:08:31,750 --> 00:08:32,850
going to be equal.

181
00:08:32,850 --> 00:08:34,909
So this is also going
to be 50 degrees.

182
00:08:34,909 --> 00:08:38,610
And that's choice B.

183
00:08:38,610 --> 00:08:40,559
Next problem.

184
00:08:40,559 --> 00:08:43,590
I don't know if I have time
for this, but I'll try.

185
00:08:43,590 --> 00:08:45,780
Problem 12.

186
00:08:45,780 --> 00:08:47,310
k is a positive integer.

187
00:08:47,310 --> 00:08:50,890
What is the least value
of k for which the

188
00:08:50,890 --> 00:08:53,310
square root of-- OK.

189
00:08:53,310 --> 00:08:59,390
So what is the least value
of k for which 5k

190
00:08:59,390 --> 00:09:02,390
over 3 is an integer.

191
00:09:02,390 --> 00:09:04,600
So this has to be a whole
number, right?

192
00:09:04,600 --> 00:09:07,720
So essentially if we want to
find the least value of k, we

193
00:09:07,720 --> 00:09:09,700
essentially want to say, well
what's the least integer that

194
00:09:09,700 --> 00:09:11,910
this could be?

195
00:09:11,910 --> 00:09:15,240
And they're telling us that
k is a positive integer.

196
00:09:15,240 --> 00:09:19,330
So first of all, in order for
the square root to be an

197
00:09:19,330 --> 00:09:24,230
integer, this whole thing has
to be an integer, right?

198
00:09:24,230 --> 00:09:27,710
So let's see, k has to
be a multiple of 3.

199
00:09:27,710 --> 00:09:31,525
In order for this expression to
be an integer, k has to be

200
00:09:31,525 --> 00:09:32,670
a multiple of 3.

201
00:09:32,670 --> 00:09:37,500
If k is 3, we get square root
of 15 over 3-- well that

202
00:09:37,500 --> 00:09:40,140
doesn't work.

203
00:09:40,140 --> 00:09:44,080
If k is 3 we just
get 5 in there.

204
00:09:44,080 --> 00:09:45,850
Actually, let me continue this
into the next problem because

205
00:09:45,850 --> 00:09:46,730
I don't want to rush this.

206
00:09:46,730 --> 00:09:48,470
I'll see you in the
next video.