[Script Info]
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[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.64,0:00:04.70,Default,,0000,0000,0000,,I got this problem here\Nfrom the 2003 AIME exam.
Dialogue: 0,0:00:04.70,0:00:08.36,Default,,0000,0000,0000,,That stands for the American\NInvitational Mathematics Exam.
Dialogue: 0,0:00:08.36,0:00:10.88,Default,,0000,0000,0000,,And this was actually the\Nfirst problem in the exam.
Dialogue: 0,0:00:10.88,0:00:17.23,Default,,0000,0000,0000,,The product N of three positive\Nintegers is 6 times their sum,
Dialogue: 0,0:00:17.23,0:00:20.06,Default,,0000,0000,0000,,and one of the integers is\Nthe sum of the other two.
Dialogue: 0,0:00:20.06,0:00:24.08,Default,,0000,0000,0000,,Find the sum of all\Npossible values of N.
Dialogue: 0,0:00:24.08,0:00:27.38,Default,,0000,0000,0000,,So we have to deal with\Nthree positive integers.
Dialogue: 0,0:00:27.38,0:00:30.65,Default,,0000,0000,0000,,So we have three positive\Nintegers right over here.
Dialogue: 0,0:00:30.65,0:00:33.08,Default,,0000,0000,0000,,So let's just think about\Nthree positive integers.
Dialogue: 0,0:00:33.08,0:00:35.40,Default,,0000,0000,0000,,Let's call them a, b, and c.
Dialogue: 0,0:00:35.40,0:00:36.28,Default,,0000,0000,0000,,They're all positive.
Dialogue: 0,0:00:36.28,0:00:37.53,Default,,0000,0000,0000,,They're all integers.
Dialogue: 0,0:00:37.53,0:00:41.16,Default,,0000,0000,0000,,The product N of these\Nthree positive integers--
Dialogue: 0,0:00:41.16,0:00:48.03,Default,,0000,0000,0000,,so a times b times c is equal\Nto N-- is 6 times their sum.
Dialogue: 0,0:00:48.03,0:00:51.11,Default,,0000,0000,0000,,This is equal to\N6 times the sum.
Dialogue: 0,0:00:51.11,0:00:52.81,Default,,0000,0000,0000,,Let me do this in another color.
Dialogue: 0,0:00:52.81,0:00:54.45,Default,,0000,0000,0000,,So this is their product.
Dialogue: 0,0:00:54.45,0:00:57.69,Default,,0000,0000,0000,,So the product N of\Nthree positive integers
Dialogue: 0,0:00:57.69,0:01:02.05,Default,,0000,0000,0000,,is 6 times their sum.
Dialogue: 0,0:01:02.05,0:01:05.31,Default,,0000,0000,0000,,So this is equal to 6\Ntimes the sum of those
Dialogue: 0,0:01:05.31,0:01:09.51,Default,,0000,0000,0000,,integers a plus b plus c.
Dialogue: 0,0:01:09.51,0:01:13.02,Default,,0000,0000,0000,,And one of the integers is\Nthe sum of the other two.
Dialogue: 0,0:01:19.95,0:01:23.44,Default,,0000,0000,0000,,Well, let's just pick c\Nto be the sum of a and b.
Dialogue: 0,0:01:23.44,0:01:24.19,Default,,0000,0000,0000,,It doesn't matter.
Dialogue: 0,0:01:24.19,0:01:26.60,Default,,0000,0000,0000,,These are just names, and\Nwe haven't said one of them
Dialogue: 0,0:01:26.60,0:01:28.33,Default,,0000,0000,0000,,is larger or less\Nthan the other one.
Dialogue: 0,0:01:28.33,0:01:31.40,Default,,0000,0000,0000,,So let's just say\Nthat a plus b is
Dialogue: 0,0:01:31.40,0:01:33.88,Default,,0000,0000,0000,,equal to c, that\None of the integers
Dialogue: 0,0:01:33.88,0:01:36.99,Default,,0000,0000,0000,,is the sum of the other two.\Nc is the sum of a plus b.
Dialogue: 0,0:01:36.99,0:01:41.99,Default,,0000,0000,0000,,Find the sum of all\Npossible values of N.
Dialogue: 0,0:01:41.99,0:01:45.16,Default,,0000,0000,0000,,So let's just try to do a\Nlittle bit of manipulation
Dialogue: 0,0:01:45.16,0:01:47.05,Default,,0000,0000,0000,,of the information we\Nhave here, and maybe we
Dialogue: 0,0:01:47.05,0:01:50.15,Default,,0000,0000,0000,,can get some relationship\Nor some constraints
Dialogue: 0,0:01:50.15,0:01:51.77,Default,,0000,0000,0000,,on our numbers, and\Nthen we can kind of
Dialogue: 0,0:01:51.77,0:01:54.05,Default,,0000,0000,0000,,go through all of\Nthe possibilities.
Dialogue: 0,0:01:54.05,0:01:56.73,Default,,0000,0000,0000,,So let's see, we know that\Na plus b is equal to c.
Dialogue: 0,0:01:56.73,0:02:02.36,Default,,0000,0000,0000,,So we can replace c\Neverywhere with a plus b.
Dialogue: 0,0:02:02.36,0:02:04.11,Default,,0000,0000,0000,,So this expression\Nright over here
Dialogue: 0,0:02:04.11,0:02:09.13,Default,,0000,0000,0000,,becomes ab, which is\Njust a times b, times c.
Dialogue: 0,0:02:09.13,0:02:11.78,Default,,0000,0000,0000,,But instead of c, I'm going to\Nwrite an a plus b over here.
Dialogue: 0,0:02:15.50,0:02:25.44,Default,,0000,0000,0000,,And then that is equal to\N6 times a plus b plus c.
Dialogue: 0,0:02:25.44,0:02:31.36,Default,,0000,0000,0000,,And so once again, I'll replace\Nwith the c with an a plus b,
Dialogue: 0,0:02:31.36,0:02:33.61,Default,,0000,0000,0000,,and then what does\Nthis simplify to?
Dialogue: 0,0:02:33.61,0:02:36.15,Default,,0000,0000,0000,,So on the right-hand side,\Nwe have 6 times a plus b
Dialogue: 0,0:02:36.15,0:02:37.02,Default,,0000,0000,0000,,plus a plus b.
Dialogue: 0,0:02:37.02,0:02:43.68,Default,,0000,0000,0000,,This is the same thing\Nas 6 times 2a plus 2b,
Dialogue: 0,0:02:43.68,0:02:45.52,Default,,0000,0000,0000,,just added the a's and the b's.
Dialogue: 0,0:02:45.52,0:02:46.70,Default,,0000,0000,0000,,And we can factor out a 2.
Dialogue: 0,0:02:46.70,0:02:49.74,Default,,0000,0000,0000,,This is the same thing as if\Nyou take out a 2, 6 times 2
Dialogue: 0,0:02:49.74,0:02:53.32,Default,,0000,0000,0000,,is 12 times a plus b.
Dialogue: 0,0:02:53.32,0:02:55.86,Default,,0000,0000,0000,,The left-hand side\Nright over here
Dialogue: 0,0:02:55.86,0:03:01.72,Default,,0000,0000,0000,,is still a times b\Nor ab times a plus b.
Dialogue: 0,0:03:01.72,0:03:07.71,Default,,0000,0000,0000,,So ab times a plus b has got to\Nbe equal to 12 times a plus b.
Dialogue: 0,0:03:07.71,0:03:09.43,Default,,0000,0000,0000,,So this is pretty\Ninteresting here.
Dialogue: 0,0:03:09.43,0:03:12.61,Default,,0000,0000,0000,,We can divide both\Nsides by a plus b.
Dialogue: 0,0:03:12.61,0:03:17.14,Default,,0000,0000,0000,,We know that a plus b cannot be\Nequal to 0 since all of these
Dialogue: 0,0:03:17.14,0:03:20.94,Default,,0000,0000,0000,,numbers have to be\Npositive numbers.
Dialogue: 0,0:03:20.94,0:03:24.74,Default,,0000,0000,0000,,And the reason why I say that\Nis if it was 0, dividing by 0
Dialogue: 0,0:03:24.74,0:03:27.45,Default,,0000,0000,0000,,would give you an\Nundefined answer.
Dialogue: 0,0:03:27.45,0:03:30.13,Default,,0000,0000,0000,,So if we divide both\Nsides by a plus b,
Dialogue: 0,0:03:30.13,0:03:34.15,Default,,0000,0000,0000,,we get a times b is equal to 12.
Dialogue: 0,0:03:34.15,0:03:36.14,Default,,0000,0000,0000,,So all the constraints\Nthat they gave us
Dialogue: 0,0:03:36.14,0:03:38.29,Default,,0000,0000,0000,,boiled down to this\Nright over here.
Dialogue: 0,0:03:38.29,0:03:41.53,Default,,0000,0000,0000,,The product of a and\Nb is equal to 12.
Dialogue: 0,0:03:41.53,0:03:43.73,Default,,0000,0000,0000,,And there's only\Nso many numbers, so
Dialogue: 0,0:03:43.73,0:03:46.10,Default,,0000,0000,0000,,many positive integers where\Nyou if you take the product,
Dialogue: 0,0:03:46.10,0:03:46.95,Default,,0000,0000,0000,,you get 12.
Dialogue: 0,0:03:46.95,0:03:49.17,Default,,0000,0000,0000,,Let's try them out.
Dialogue: 0,0:03:49.17,0:03:50.59,Default,,0000,0000,0000,,So let me write\Nsome columns here.
Dialogue: 0,0:03:50.59,0:03:54.29,Default,,0000,0000,0000,,Let's say a, b, c.
Dialogue: 0,0:03:54.29,0:04:00.07,Default,,0000,0000,0000,,And then we care\Nabout their product,
Dialogue: 0,0:04:00.07,0:04:03.70,Default,,0000,0000,0000,,so I'll write that\Nover here, so abc.
Dialogue: 0,0:04:03.70,0:04:08.05,Default,,0000,0000,0000,,So if a is 1, b\Nis going to be 12.
Dialogue: 0,0:04:08.05,0:04:13.53,Default,,0000,0000,0000,,c is the sum of those two, so\Nc is going to be 13, 1 times
Dialogue: 0,0:04:13.53,0:04:15.38,Default,,0000,0000,0000,,12 times 13.
Dialogue: 0,0:04:15.38,0:04:21.96,Default,,0000,0000,0000,,12 times 12 is 144 plus\Nanother 12 is going to be 156.
Dialogue: 0,0:04:21.96,0:04:24.62,Default,,0000,0000,0000,,And just for fun, you\Ncan verify that this
Dialogue: 0,0:04:24.62,0:04:27.04,Default,,0000,0000,0000,,is going to be equal\Nto 6 times their sum.
Dialogue: 0,0:04:27.04,0:04:32.28,Default,,0000,0000,0000,,Their sum is 26,\N26 times 6 is 156.
Dialogue: 0,0:04:32.28,0:04:33.53,Default,,0000,0000,0000,,So this one definitely worked.
Dialogue: 0,0:04:33.53,0:04:34.85,Default,,0000,0000,0000,,It definitely worked\Nfor the constraints.
Dialogue: 0,0:04:34.85,0:04:37.10,Default,,0000,0000,0000,,And it should because we\Nboiled down those constraints
Dialogue: 0,0:04:37.10,0:04:40.06,Default,,0000,0000,0000,,to a times b need\Nto be equal to 12.
Dialogue: 0,0:04:40.06,0:04:41.72,Default,,0000,0000,0000,,So let's try another one.
Dialogue: 0,0:04:41.72,0:04:45.67,Default,,0000,0000,0000,,2 times 6, their sum is 8.
Dialogue: 0,0:04:45.67,0:04:48.20,Default,,0000,0000,0000,,And then if I were to take\Nthe product of all of these,
Dialogue: 0,0:04:48.20,0:04:52.73,Default,,0000,0000,0000,,you get 2 times 6\Nis 12 times 8 is 96.
Dialogue: 0,0:04:55.37,0:04:58.83,Default,,0000,0000,0000,,And then we could try 3 and 4.
Dialogue: 0,0:04:58.83,0:05:01.15,Default,,0000,0000,0000,,3 plus 4 is 7.
Dialogue: 0,0:05:01.15,0:05:06.70,Default,,0000,0000,0000,,3 times 4 is 12 times 7.
Dialogue: 0,0:05:06.70,0:05:09.08,Default,,0000,0000,0000,,Actually, I should have known\Nthe a times b is always 12,
Dialogue: 0,0:05:09.08,0:05:11.56,Default,,0000,0000,0000,,so you just have to multiply\N12 times this last column.
Dialogue: 0,0:05:11.56,0:05:14.19,Default,,0000,0000,0000,,12 times 7 is 84.
Dialogue: 0,0:05:17.08,0:05:19.20,Default,,0000,0000,0000,,And there aren't any others.
Dialogue: 0,0:05:19.20,0:05:21.24,Default,,0000,0000,0000,,You definitely can't go\Nabove 12 because then you
Dialogue: 0,0:05:21.24,0:05:22.58,Default,,0000,0000,0000,,would have to deal\Nwith non-integers.
Dialogue: 0,0:05:22.58,0:05:23.84,Default,,0000,0000,0000,,You would have to\Ndeal with fractions.
Dialogue: 0,0:05:23.84,0:05:25.52,Default,,0000,0000,0000,,You can't do the negative\Nversions of these
Dialogue: 0,0:05:25.52,0:05:27.44,Default,,0000,0000,0000,,because they all have\Nto be positive integers.
Dialogue: 0,0:05:27.44,0:05:28.12,Default,,0000,0000,0000,,So that's it.
Dialogue: 0,0:05:28.12,0:05:30.62,Default,,0000,0000,0000,,Those are all of the\Npossible positive integers
Dialogue: 0,0:05:30.62,0:05:33.07,Default,,0000,0000,0000,,where if you take their\Nproducts you get 12.
Dialogue: 0,0:05:33.07,0:05:35.30,Default,,0000,0000,0000,,We've essentially\Njust factored 12.
Dialogue: 0,0:05:35.30,0:05:41.62,Default,,0000,0000,0000,,So they want us to find the sum\Nof all possible values of N.
Dialogue: 0,0:05:41.62,0:05:43.42,Default,,0000,0000,0000,,Well, these are all\Nthe possible values
Dialogue: 0,0:05:43.42,0:05:45.35,Default,,0000,0000,0000,,of N. N was the product\Nof those integers.
Dialogue: 0,0:05:45.35,0:05:47.92,Default,,0000,0000,0000,,So let's just take the sum.
Dialogue: 0,0:05:47.92,0:05:52.98,Default,,0000,0000,0000,,6 plus 6 is 12 plus\N4 is 16, 1 plus 5
Dialogue: 0,0:05:52.98,0:06:01.74,Default,,0000,0000,0000,,is 6 plus 9 is 15 plus\N8 is 23, 2 plus 1 is 3.
Dialogue: 0,0:06:01.74,0:06:04.75,Default,,0000,0000,0000,,So our answer is 336.