[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.70,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.70,0:00:03.13,Default,,0000,0000,0000,,William and Luis are in\Ndifferent physics classes
Dialogue: 0,0:00:03.13,0:00:04.37,Default,,0000,0000,0000,,at Santa Rita.
Dialogue: 0,0:00:04.37,0:00:07.75,Default,,0000,0000,0000,,Luis's teacher always gives\Nexams with 30 questions
Dialogue: 0,0:00:07.75,0:00:10.87,Default,,0000,0000,0000,,on them, while William's\Nteacher gives more
Dialogue: 0,0:00:10.87,0:00:14.15,Default,,0000,0000,0000,,frequent exams with\Nonly 24 questions.
Dialogue: 0,0:00:14.15,0:00:17.80,Default,,0000,0000,0000,,Luis's teacher also assigns\Nthree projects per year.
Dialogue: 0,0:00:17.80,0:00:20.26,Default,,0000,0000,0000,,Even though the two classes\Nhave to take a different number
Dialogue: 0,0:00:20.26,0:00:22.27,Default,,0000,0000,0000,,of exams, their\Nteachers have told them
Dialogue: 0,0:00:22.27,0:00:25.25,Default,,0000,0000,0000,,that both classes-- let me\Nunderline-- both classes will
Dialogue: 0,0:00:25.25,0:00:29.04,Default,,0000,0000,0000,,get the same total number\Nof exam questions each year.
Dialogue: 0,0:00:29.04,0:00:32.85,Default,,0000,0000,0000,,What is the minimum\Nnumber of exam questions
Dialogue: 0,0:00:32.85,0:00:36.81,Default,,0000,0000,0000,,William's or Luis's class can\Nexpect to get in a given year?
Dialogue: 0,0:00:36.81,0:00:38.39,Default,,0000,0000,0000,,So let's think about\Nwhat's happening.
Dialogue: 0,0:00:38.39,0:00:40.01,Default,,0000,0000,0000,,So if we think about\NLuis's teacher who
Dialogue: 0,0:00:40.01,0:00:44.59,Default,,0000,0000,0000,,gives 30 questions per test,\Nso after the first test,
Dialogue: 0,0:00:44.59,0:00:46.85,Default,,0000,0000,0000,,he would have done 30 questions.
Dialogue: 0,0:00:46.85,0:00:48.75,Default,,0000,0000,0000,,So this is 0 right over here.
Dialogue: 0,0:00:48.75,0:00:52.24,Default,,0000,0000,0000,,Then after the second test,\Nhe would have done 60.
Dialogue: 0,0:00:52.24,0:00:56.15,Default,,0000,0000,0000,,Then after the third test,\Nhe would have done 90.
Dialogue: 0,0:00:56.15,0:01:00.07,Default,,0000,0000,0000,,And after the fourth test,\Nhe would have done 120.
Dialogue: 0,0:01:00.07,0:01:03.48,Default,,0000,0000,0000,,And after the fifth test,\Nif there is a fifth test,
Dialogue: 0,0:01:03.48,0:01:06.70,Default,,0000,0000,0000,,he would do-- so this is if\Nthey have that many tests-- he
Dialogue: 0,0:01:06.70,0:01:08.91,Default,,0000,0000,0000,,would get to 150\Ntotal questions.
Dialogue: 0,0:01:08.91,0:01:10.62,Default,,0000,0000,0000,,And we could keep\Ngoing on and on looking
Dialogue: 0,0:01:10.62,0:01:12.47,Default,,0000,0000,0000,,at all the multiples of 30.
Dialogue: 0,0:01:12.47,0:01:14.80,Default,,0000,0000,0000,,So this is probably a hint\Nof what we're thinking about.
Dialogue: 0,0:01:14.80,0:01:16.55,Default,,0000,0000,0000,,We're looking at\Nmultiples of the numbers.
Dialogue: 0,0:01:16.55,0:01:19.71,Default,,0000,0000,0000,,We want the minimum multiples\Nor the least multiple.
Dialogue: 0,0:01:19.71,0:01:20.95,Default,,0000,0000,0000,,So that's with Luis.
Dialogue: 0,0:01:20.95,0:01:22.71,Default,,0000,0000,0000,,Well what's going\Non with William?
Dialogue: 0,0:01:22.71,0:01:25.65,Default,,0000,0000,0000,,Will William's teacher,\Nafter the first test,
Dialogue: 0,0:01:25.65,0:01:29.22,Default,,0000,0000,0000,,they're going to\Nget to 24 questions.
Dialogue: 0,0:01:29.22,0:01:32.77,Default,,0000,0000,0000,,Then they're going to get\Nto 48 after the second test.
Dialogue: 0,0:01:32.77,0:01:37.42,Default,,0000,0000,0000,,Then they're going to get\Nto 72 after the third test.
Dialogue: 0,0:01:37.42,0:01:39.25,Default,,0000,0000,0000,,Then they're going to get to 96.
Dialogue: 0,0:01:39.25,0:01:41.82,Default,,0000,0000,0000,,I'm just taking multiples of 24.
Dialogue: 0,0:01:41.82,0:01:45.03,Default,,0000,0000,0000,,They're going to get to\N96 after the fourth test.
Dialogue: 0,0:01:45.03,0:01:49.61,Default,,0000,0000,0000,,And then after the fifth test,\Nthey're going to get to 120.
Dialogue: 0,0:01:49.61,0:01:55.16,Default,,0000,0000,0000,,And if there's a sixth test,\Nthen they would get to 144.
Dialogue: 0,0:01:55.16,0:01:57.43,Default,,0000,0000,0000,,And we could keep going\Non and on in there.
Dialogue: 0,0:01:57.43,0:01:58.30,Default,,0000,0000,0000,,But let's see what\Nthey're asking us.
Dialogue: 0,0:01:58.30,0:02:00.18,Default,,0000,0000,0000,,What is the minimum\Nnumber of exam questions
Dialogue: 0,0:02:00.18,0:02:03.20,Default,,0000,0000,0000,,William's or Luis's class\Ncan expect to get in a year?
Dialogue: 0,0:02:03.20,0:02:04.71,Default,,0000,0000,0000,,Well the minimum\Nnumber is the point
Dialogue: 0,0:02:04.71,0:02:07.38,Default,,0000,0000,0000,,at which they've gotten the\Nsame number of exam questions,
Dialogue: 0,0:02:07.38,0:02:09.19,Default,,0000,0000,0000,,despite the fact\Nthat the tests had
Dialogue: 0,0:02:09.19,0:02:10.62,Default,,0000,0000,0000,,a different number of items.
Dialogue: 0,0:02:10.62,0:02:12.95,Default,,0000,0000,0000,,And you see the point at which\Nthey have the same number
Dialogue: 0,0:02:12.95,0:02:14.88,Default,,0000,0000,0000,,is at 120.
Dialogue: 0,0:02:14.88,0:02:16.77,Default,,0000,0000,0000,,This happens at 120.
Dialogue: 0,0:02:16.77,0:02:19.30,Default,,0000,0000,0000,,They both could have\Nexactly 120 questions
Dialogue: 0,0:02:19.30,0:02:21.84,Default,,0000,0000,0000,,even though Luis's teacher\Nis giving 30 at a time
Dialogue: 0,0:02:21.84,0:02:25.24,Default,,0000,0000,0000,,and even though William's\Nteacher is giving 24 at a time.
Dialogue: 0,0:02:25.24,0:02:28.47,Default,,0000,0000,0000,,And so the answer is 120.
Dialogue: 0,0:02:28.47,0:02:30.51,Default,,0000,0000,0000,,And notice, they had a\Ndifferent number of exams.
Dialogue: 0,0:02:30.51,0:02:33.65,Default,,0000,0000,0000,,Luis had one, two,\Nthree, four exams
Dialogue: 0,0:02:33.65,0:02:36.30,Default,,0000,0000,0000,,while William would have to\Nhave one, two, three, four,
Dialogue: 0,0:02:36.30,0:02:37.57,Default,,0000,0000,0000,,five exams.
Dialogue: 0,0:02:37.57,0:02:41.27,Default,,0000,0000,0000,,But that gets them both\Nto 120 total questions.
Dialogue: 0,0:02:41.27,0:02:44.10,Default,,0000,0000,0000,,Now thinking of it in terms\Nof some of the math notation
Dialogue: 0,0:02:44.10,0:02:47.37,Default,,0000,0000,0000,,or the least common multiple\Nnotation we've seen before,
Dialogue: 0,0:02:47.37,0:02:55.65,Default,,0000,0000,0000,,this is really asking us what is\Nthe least common multiple of 30
Dialogue: 0,0:02:55.65,0:02:56.98,Default,,0000,0000,0000,,and 24.
Dialogue: 0,0:02:56.98,0:03:02.69,Default,,0000,0000,0000,,And that least common\Nmultiple is equal to 120.
Dialogue: 0,0:03:02.69,0:03:04.15,Default,,0000,0000,0000,,Now there's other\Nways that you can
Dialogue: 0,0:03:04.15,0:03:06.40,Default,,0000,0000,0000,,find the least common multiple\Nother than just looking
Dialogue: 0,0:03:06.40,0:03:07.87,Default,,0000,0000,0000,,at the multiples like this.
Dialogue: 0,0:03:07.87,0:03:10.44,Default,,0000,0000,0000,,You could look at it\Nthrough prime factorization.
Dialogue: 0,0:03:10.44,0:03:15.29,Default,,0000,0000,0000,,30 is 2 times 15,\Nwhich is 3 times 5.
Dialogue: 0,0:03:15.29,0:03:20.42,Default,,0000,0000,0000,,So we could say that 30 is\Nequal to 2 times 3 times 5.
Dialogue: 0,0:03:20.42,0:03:28.58,Default,,0000,0000,0000,,And 24-- that's a different\Ncolor than that blue-- 24
Dialogue: 0,0:03:28.58,0:03:31.57,Default,,0000,0000,0000,,is equal to 2 times 12.
Dialogue: 0,0:03:31.57,0:03:33.85,Default,,0000,0000,0000,,12 is equal to 2 times 6.
Dialogue: 0,0:03:33.85,0:03:36.08,Default,,0000,0000,0000,,6 is equal to 2 times 3.
Dialogue: 0,0:03:36.08,0:03:44.66,Default,,0000,0000,0000,,So 24 is equal to 2\Ntimes 2 times 2 times 3.
Dialogue: 0,0:03:44.66,0:03:47.25,Default,,0000,0000,0000,,So another way to come up with\Nthe least common multiple,
Dialogue: 0,0:03:47.25,0:03:49.72,Default,,0000,0000,0000,,if we didn't even do this\Nexercise up here, says, look,
Dialogue: 0,0:03:49.72,0:03:52.82,Default,,0000,0000,0000,,the number has to be\Ndivisible by both 30 and 24.
Dialogue: 0,0:03:52.82,0:03:54.81,Default,,0000,0000,0000,,If it's going to\Nbe divisible by 30,
Dialogue: 0,0:03:54.81,0:04:00.06,Default,,0000,0000,0000,,it's going to have to\Nhave 2 times 3 times 5
Dialogue: 0,0:04:00.06,0:04:01.43,Default,,0000,0000,0000,,in its prime factorization.
Dialogue: 0,0:04:01.43,0:04:03.42,Default,,0000,0000,0000,,That is essentially 30.
Dialogue: 0,0:04:03.42,0:04:05.83,Default,,0000,0000,0000,,So this makes it\Ndivisible by 30.
Dialogue: 0,0:04:05.83,0:04:10.05,Default,,0000,0000,0000,,And say, well in order\Nto be divisible by 24,
Dialogue: 0,0:04:10.05,0:04:13.75,Default,,0000,0000,0000,,its prime factorization is\Ngoing to need 3 twos and a 3.
Dialogue: 0,0:04:13.75,0:04:15.23,Default,,0000,0000,0000,,Well we already have 1 three.
Dialogue: 0,0:04:15.23,0:04:18.04,Default,,0000,0000,0000,,And we already have 1 two,\Nso we just need 2 more twos.
Dialogue: 0,0:04:18.04,0:04:20.74,Default,,0000,0000,0000,,So 2 times 2.
Dialogue: 0,0:04:20.74,0:04:24.34,Default,,0000,0000,0000,,So this makes it--\Nlet me scroll up
Dialogue: 0,0:04:24.34,0:04:29.08,Default,,0000,0000,0000,,a little bit-- this right over\Nhere makes it divisible by 24.
Dialogue: 0,0:04:29.08,0:04:32.03,Default,,0000,0000,0000,,And so this is essentially\Nthe prime factorization
Dialogue: 0,0:04:32.03,0:04:34.92,Default,,0000,0000,0000,,of the least common\Nmultiple of 30 and 24.
Dialogue: 0,0:04:34.92,0:04:37.30,Default,,0000,0000,0000,,You take any one of\Nthese numbers away,
Dialogue: 0,0:04:37.30,0:04:40.25,Default,,0000,0000,0000,,you are no longer going to be\Ndivisible by one of these two
Dialogue: 0,0:04:40.25,0:04:40.75,Default,,0000,0000,0000,,numbers.
Dialogue: 0,0:04:40.75,0:04:43.33,Default,,0000,0000,0000,,If you take a two away, you're\Nnot going to be divisible by 24
Dialogue: 0,0:04:43.33,0:04:43.95,Default,,0000,0000,0000,,anymore.
Dialogue: 0,0:04:43.95,0:04:45.83,Default,,0000,0000,0000,,If you take a two\Nor a three away.
Dialogue: 0,0:04:45.83,0:04:50.52,Default,,0000,0000,0000,,If you take a three\Nor a five away,
Dialogue: 0,0:04:50.52,0:04:53.14,Default,,0000,0000,0000,,you're not going to be\Ndivisible by 30 anymore.
Dialogue: 0,0:04:53.14,0:04:55.02,Default,,0000,0000,0000,,And so if you were to\Nmultiply all these out,
Dialogue: 0,0:04:55.02,0:05:04.17,Default,,0000,0000,0000,,this is 2 times 2 times 2 is 8\Ntimes 3 is 24 times 5 is 120.
Dialogue: 0,0:05:04.17,0:05:06.74,Default,,0000,0000,0000,,Now let's do one more of these.
Dialogue: 0,0:05:06.74,0:05:09.97,Default,,0000,0000,0000,,Umama just bought one\Npackage of 21 binders.
Dialogue: 0,0:05:09.97,0:05:11.22,Default,,0000,0000,0000,,Let me write that number down.
Dialogue: 0,0:05:11.22,0:05:12.66,Default,,0000,0000,0000,,21 binders.
Dialogue: 0,0:05:12.66,0:05:14.80,Default,,0000,0000,0000,,She also bought a\Npackage of 30 pencils.
Dialogue: 0,0:05:14.80,0:05:17.86,Default,,0000,0000,0000,,
Dialogue: 0,0:05:17.86,0:05:20.24,Default,,0000,0000,0000,,She wants to use all of\Nthe binders and pencils
Dialogue: 0,0:05:20.24,0:05:23.06,Default,,0000,0000,0000,,to create identical\Nsets of office supplies
Dialogue: 0,0:05:23.06,0:05:24.65,Default,,0000,0000,0000,,for her classmates.
Dialogue: 0,0:05:24.65,0:05:27.54,Default,,0000,0000,0000,,What is the greatest\Nnumber of identical sets
Dialogue: 0,0:05:27.54,0:05:29.46,Default,,0000,0000,0000,,Umama can make using\Nall the supplies?
Dialogue: 0,0:05:29.46,0:05:31.33,Default,,0000,0000,0000,,So the fact that we're\Ntalking about greatest
Dialogue: 0,0:05:31.33,0:05:33.25,Default,,0000,0000,0000,,is clue that it's probably\Ngoing to be dealing
Dialogue: 0,0:05:33.25,0:05:34.62,Default,,0000,0000,0000,,with greatest common divisors.
Dialogue: 0,0:05:34.62,0:05:36.71,Default,,0000,0000,0000,,And it's also dealing with\Ndividing these things.
Dialogue: 0,0:05:36.71,0:05:39.66,Default,,0000,0000,0000,,We want to divide these\Nboth into the greatest
Dialogue: 0,0:05:39.66,0:05:44.76,Default,,0000,0000,0000,,number of identical sets.
Dialogue: 0,0:05:44.76,0:05:46.93,Default,,0000,0000,0000,,So there's a couple of ways\Nwe could think about it.
Dialogue: 0,0:05:46.93,0:05:49.06,Default,,0000,0000,0000,,Let's think about what the\Ngreatest common divisor
Dialogue: 0,0:05:49.06,0:05:51.10,Default,,0000,0000,0000,,of both these numbers are.
Dialogue: 0,0:05:51.10,0:05:53.45,Default,,0000,0000,0000,,Or I could even say the\Ngreatest common factor.
Dialogue: 0,0:05:53.45,0:06:00.50,Default,,0000,0000,0000,,The greatest common\Ndivisor of 21 and 30.
Dialogue: 0,0:06:00.50,0:06:04.28,Default,,0000,0000,0000,,So what's the largest number\Nthat divides into both of them?
Dialogue: 0,0:06:04.28,0:06:05.90,Default,,0000,0000,0000,,So we could go with\Nthe prime factor.
Dialogue: 0,0:06:05.90,0:06:07.61,Default,,0000,0000,0000,,We could list all of\Ntheir normal factors
Dialogue: 0,0:06:07.61,0:06:09.57,Default,,0000,0000,0000,,and see what is the\Ngreatest common one.
Dialogue: 0,0:06:09.57,0:06:16.70,Default,,0000,0000,0000,,Or we could look at the\Nprime factorization.
Dialogue: 0,0:06:16.70,0:06:18.82,Default,,0000,0000,0000,,So let's just do the prime\Nfactorization method.
Dialogue: 0,0:06:18.82,0:06:21.76,Default,,0000,0000,0000,,So 21 is the same\Nthing as 3 times 7.
Dialogue: 0,0:06:21.76,0:06:23.69,Default,,0000,0000,0000,,These are both prime numbers.
Dialogue: 0,0:06:23.69,0:06:27.14,Default,,0000,0000,0000,,30 is, let's see,\Nit's 3-- actually,
Dialogue: 0,0:06:27.14,0:06:30.21,Default,,0000,0000,0000,,I could write it this\Nway-- it is 2 times 15.
Dialogue: 0,0:06:30.21,0:06:32.11,Default,,0000,0000,0000,,We already did it\Nactually just now.
Dialogue: 0,0:06:32.11,0:06:34.62,Default,,0000,0000,0000,,And 15 is 3 times 5.
Dialogue: 0,0:06:34.62,0:06:37.68,Default,,0000,0000,0000,,So what's the largest\Nnumber of prime numbers that
Dialogue: 0,0:06:37.68,0:06:39.78,Default,,0000,0000,0000,,are common to both\Nfactorizations?
Dialogue: 0,0:06:39.78,0:06:42.82,Default,,0000,0000,0000,,Well you only have a\Nthree right over here.
Dialogue: 0,0:06:42.82,0:06:44.82,Default,,0000,0000,0000,,Then you don't have a\Nthree times anything else.
Dialogue: 0,0:06:44.82,0:06:47.42,Default,,0000,0000,0000,,So this is just going\Nto be equal to 3.
Dialogue: 0,0:06:47.42,0:06:48.90,Default,,0000,0000,0000,,So this is essentially\Ntelling us,
Dialogue: 0,0:06:48.90,0:06:54.76,Default,,0000,0000,0000,,look, we can divide both\Nof these numbers into 3
Dialogue: 0,0:06:54.76,0:06:56.74,Default,,0000,0000,0000,,and that will give\Nus the largest
Dialogue: 0,0:06:56.74,0:06:58.50,Default,,0000,0000,0000,,number of identical sets.
Dialogue: 0,0:06:58.50,0:07:00.17,Default,,0000,0000,0000,,So just to be clear\Nof what we're doing.
Dialogue: 0,0:07:00.17,0:07:02.26,Default,,0000,0000,0000,,So we've answered\Nthe question is 3,
Dialogue: 0,0:07:02.26,0:07:04.36,Default,,0000,0000,0000,,but just to visualize\Nit for this question,
Dialogue: 0,0:07:04.36,0:07:07.07,Default,,0000,0000,0000,,let's actually draw 21 binders.
Dialogue: 0,0:07:07.07,0:07:13.73,Default,,0000,0000,0000,,So let's say the 21 binders so\N1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
Dialogue: 0,0:07:13.73,0:07:19.32,Default,,0000,0000,0000,,11, 12, 13, 14, 15,\N16, 17, 18, 19, 20, 21.
Dialogue: 0,0:07:19.32,0:07:22.76,Default,,0000,0000,0000,,And then 30 pencils, so\NI'll just do those in green.
Dialogue: 0,0:07:22.76,0:07:27.70,Default,,0000,0000,0000,,So 1, 2, 3, 4, 5,\N6, 7, 8, 9, 10.
Dialogue: 0,0:07:27.70,0:07:29.48,Default,,0000,0000,0000,,Let me just copy and paste that.
Dialogue: 0,0:07:29.48,0:07:31.66,Default,,0000,0000,0000,,This is getting tedious.
Dialogue: 0,0:07:31.66,0:07:35.51,Default,,0000,0000,0000,,So copy and paste.
Dialogue: 0,0:07:35.51,0:07:41.63,Default,,0000,0000,0000,,So that's 20 and then\Npaste that is 30.
Dialogue: 0,0:07:41.63,0:07:45.03,Default,,0000,0000,0000,,Now, we figured out that 3\Nis the largest number that
Dialogue: 0,0:07:45.03,0:07:46.75,Default,,0000,0000,0000,,divides into both\Nof these evenly.
Dialogue: 0,0:07:46.75,0:07:50.67,Default,,0000,0000,0000,,So I can divide both of\Nthese into groups of 3.
Dialogue: 0,0:07:50.67,0:07:55.39,Default,,0000,0000,0000,,So for the binders, I could\Ndo it into three groups of 7.
Dialogue: 0,0:07:55.39,0:07:58.40,Default,,0000,0000,0000,,And then for the\Npencils, I could do it
Dialogue: 0,0:07:58.40,0:08:01.32,Default,,0000,0000,0000,,into three groups of 10.
Dialogue: 0,0:08:01.32,0:08:03.05,Default,,0000,0000,0000,,So if there are\Nthree people that
Dialogue: 0,0:08:03.05,0:08:05.71,Default,,0000,0000,0000,,are coming into\Nthis classroom, I
Dialogue: 0,0:08:05.71,0:08:11.64,Default,,0000,0000,0000,,could give them each seven\Nbinders and 10 pencils.
Dialogue: 0,0:08:11.64,0:08:14.29,Default,,0000,0000,0000,,But that's the greatest\Nnumber of identical sets
Dialogue: 0,0:08:14.29,0:08:15.27,Default,,0000,0000,0000,,Umama can make.
Dialogue: 0,0:08:15.27,0:08:16.45,Default,,0000,0000,0000,,I would have three sets.
Dialogue: 0,0:08:16.45,0:08:22.00,Default,,0000,0000,0000,,Each set would have seven\Nbinders and 10 pencils.
Dialogue: 0,0:08:22.00,0:08:23.50,Default,,0000,0000,0000,,And we essentially\Nare just thinking
Dialogue: 0,0:08:23.50,0:08:27.96,Default,,0000,0000,0000,,about what's the number that we\Ncan divide both of these sets
Dialogue: 0,0:08:27.96,0:08:30.05,Default,,0000,0000,0000,,into evenly, the\Nlargest number that we
Dialogue: 0,0:08:30.05,0:08:33.26,Default,,0000,0000,0000,,can divide both of\Nthese sets into evenly.
Dialogue: 0,0:08:33.26,0:08:33.76,Default,,0000,0000,0000,,