[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.73,0:00:04.70,Default,,0000,0000,0000,,I got this problem here from the 2003 AIME\Nexam.
Dialogue: 0,0:00:04.70,0:00:08.34,Default,,0000,0000,0000,,That stands for the American Invitational\NMathematics Exam, and
Dialogue: 0,0:00:08.34,0:00:10.32,Default,,0000,0000,0000,,this was actually the first problem in the\Nexam.
Dialogue: 0,0:00:10.32,0:00:17.33,Default,,0000,0000,0000,,The product N of three positive integers\Nis six times their sum,
Dialogue: 0,0:00:17.33,0:00:20.18,Default,,0000,0000,0000,,and one of the integers is the sum of the\Nother two.
Dialogue: 0,0:00:20.18,0:00:24.16,Default,,0000,0000,0000,,Find the sum of all possible values of N.
Dialogue: 0,0:00:24.16,0:00:27.40,Default,,0000,0000,0000,,So we have to deal with three positive\Nintegers.
Dialogue: 0,0:00:27.40,0:00:30.42,Default,,0000,0000,0000,,So we have three positive integers right\Nover
Dialogue: 0,0:00:30.42,0:00:32.91,Default,,0000,0000,0000,,here, so let's just think about three\Npositive integers.
Dialogue: 0,0:00:32.91,0:00:35.45,Default,,0000,0000,0000,,Let's call them a, b and c.
Dialogue: 0,0:00:35.45,0:00:37.67,Default,,0000,0000,0000,,They're all positive, they're all\Nintegers.
Dialogue: 0,0:00:37.67,0:00:41.11,Default,,0000,0000,0000,,The product N, of these 3 positive, these\N3 positive integers.
Dialogue: 0,0:00:41.11,0:00:48.12,Default,,0000,0000,0000,,So a times b times c is equal to N, is 6\Ntimes their sum.
Dialogue: 0,0:00:48.12,0:00:50.66,Default,,0000,0000,0000,,This is equal to 6 times the sum.
Dialogue: 0,0:00:50.66,0:00:52.84,Default,,0000,0000,0000,,Let me do this in another color.
Dialogue: 0,0:00:52.84,0:00:54.43,Default,,0000,0000,0000,,So, this is their product.
Dialogue: 0,0:00:54.43,0:01:02.08,Default,,0000,0000,0000,,So, the product N of three positive\Nintegers is 6 times, is 6 times their sum.
Dialogue: 0,0:01:02.08,0:01:09.65,Default,,0000,0000,0000,,So, this is equal to six times the sum of\Nthose integers, a plus b plus c.
Dialogue: 0,0:01:09.65,0:01:12.98,Default,,0000,0000,0000,,And one of the integers is the sum of the\Nother two.
Dialogue: 0,0:01:14.17,0:01:19.24,Default,,0000,0000,0000,,One, one of the integers is the sum of the\Nother two.
Dialogue: 0,0:01:19.24,0:01:22.66,Default,,0000,0000,0000,,Well let's just pick c to be the sum of a\Nand b.
Dialogue: 0,0:01:22.66,0:01:25.60,Default,,0000,0000,0000,,We could do, it doesn't matter, these are\Njust names and we
Dialogue: 0,0:01:25.60,0:01:28.27,Default,,0000,0000,0000,,haven't said one of them is larger or less\Nthan the other one.
Dialogue: 0,0:01:28.27,0:01:32.22,Default,,0000,0000,0000,,So let's just said that a plus b is equal\Nto c.
Dialogue: 0,0:01:32.22,0:01:37.09,Default,,0000,0000,0000,,The one of the integers is the sum of the\Nother two, c is the sum of a plus b.
Dialogue: 0,0:01:37.09,0:01:40.91,Default,,0000,0000,0000,,Find the sum of all possible values of N.
Dialogue: 0,0:01:40.91,0:01:44.42,Default,,0000,0000,0000,,So let's just try to do a little bit of
Dialogue: 0,0:01:44.42,0:01:47.26,Default,,0000,0000,0000,,manipulation of the information of what we\Nhave here and maybe,
Dialogue: 0,0:01:47.26,0:01:50.57,Default,,0000,0000,0000,,we can get some relationship or some\Nconstraints on our
Dialogue: 0,0:01:50.57,0:01:54.05,Default,,0000,0000,0000,,numbers and then we can kinda go through\Nall the possibilities.
Dialogue: 0,0:01:54.05,0:01:56.73,Default,,0000,0000,0000,,So let's see, we know that a plus b is\Nequal to c.
Dialogue: 0,0:01:56.73,0:02:02.97,Default,,0000,0000,0000,,So we could replace c, we can replace c\Neverywhere with a plus b, so this
Dialogue: 0,0:02:02.97,0:02:09.22,Default,,0000,0000,0000,,expression right over here becomes ab,\Nwhich is just a times b, times c, but
Dialogue: 0,0:02:09.22,0:02:14.36,Default,,0000,0000,0000,,instead of c, I'm gonna write an a plus b\Nover here, a plus b,
Dialogue: 0,0:02:15.62,0:02:22.14,Default,,0000,0000,0000,,and then that is equal to 6 times, that is\Nequal to 6 times a plus b,
Dialogue: 0,0:02:22.14,0:02:24.96,Default,,0000,0000,0000,,a plus b plus c.
Dialogue: 0,0:02:24.96,0:02:30.79,Default,,0000,0000,0000,,And so, once again I'll replace the c with\Nan a plus b.
Dialogue: 0,0:02:30.79,0:02:33.71,Default,,0000,0000,0000,,And then what does this simplify to.
Dialogue: 0,0:02:33.71,0:02:37.02,Default,,0000,0000,0000,,So on the right hand side, we have 6 time\Na plus b plus a plus b.
Dialogue: 0,0:02:37.02,0:02:43.69,Default,,0000,0000,0000,,This is the same thing as 6 times 2a plus\N2b, 2a plus 2b,
Dialogue: 0,0:02:43.69,0:02:46.82,Default,,0000,0000,0000,,just added the a's and the b's and we can\Nfactor out a 2.
Dialogue: 0,0:02:46.82,0:02:52.35,Default,,0000,0000,0000,,This is the same thing as if you take out\Na 2, 6 times 2 is 12 times a
Dialogue: 0,0:02:52.35,0:02:57.44,Default,,0000,0000,0000,,plus b, the left hand side over here is\Nstill, is still a
Dialogue: 0,0:02:57.44,0:03:02.43,Default,,0000,0000,0000,,times b, or a b, times a plus b, so ab\Ntimes
Dialogue: 0,0:03:02.43,0:03:07.81,Default,,0000,0000,0000,,a plus b has got to be equal to 12 times a\Nplus b.
Dialogue: 0,0:03:07.81,0:03:12.68,Default,,0000,0000,0000,,So this is pretty interesting here, we can\Ndivide both sides by a plus b.
Dialogue: 0,0:03:12.68,0:03:15.74,Default,,0000,0000,0000,,We know that a plus b won't be equal to,\Ncannot be
Dialogue: 0,0:03:15.74,0:03:19.48,Default,,0000,0000,0000,,equal to zero since all of these numbers\Nhave to be positive numbers.
Dialogue: 0,0:03:19.48,0:03:22.06,Default,,0000,0000,0000,,So if we divide both sides, and the reason\Nwhy I say that is you,
Dialogue: 0,0:03:22.06,0:03:27.45,Default,,0000,0000,0000,,if you divide, if it was zero, dividing by\Nzero would give you an undefined answer.
Dialogue: 0,0:03:27.45,0:03:34.13,Default,,0000,0000,0000,,So if we divide both sides by a plus b, we\Nget a times b is equal to twelve.
Dialogue: 0,0:03:34.13,0:03:37.43,Default,,0000,0000,0000,,So all the constraints that they gave us\Nboiled down to
Dialogue: 0,0:03:37.43,0:03:40.59,Default,,0000,0000,0000,,this right over here, the product of a and\Nb is
Dialogue: 0,0:03:40.59,0:03:43.99,Default,,0000,0000,0000,,equal to 12 and there's only so many\Nnumbers, so many
Dialogue: 0,0:03:43.99,0:03:46.90,Default,,0000,0000,0000,,positive integers where, if you take their\Nproduct, you get twelve.
Dialogue: 0,0:03:46.90,0:03:48.52,Default,,0000,0000,0000,,Let's try them out.
Dialogue: 0,0:03:48.52,0:03:49.52,Default,,0000,0000,0000,,Let's try them out.
Dialogue: 0,0:03:49.52,0:03:50.62,Default,,0000,0000,0000,,So let me try some columns here.
Dialogue: 0,0:03:50.62,0:03:58.73,Default,,0000,0000,0000,,Let's say a, b, c, and then we care, we\Ncare about their product.
Dialogue: 0,0:03:58.73,0:03:59.98,Default,,0000,0000,0000,,We care about their product.
Dialogue: 0,0:03:59.98,0:04:01.11,Default,,0000,0000,0000,,So I'll write that over here.
Dialogue: 0,0:04:01.11,0:04:03.80,Default,,0000,0000,0000,,So a, b, c.
Dialogue: 0,0:04:03.80,0:04:09.77,Default,,0000,0000,0000,,So if a is 1, if a is 1, b is going to be\N12, c is the sum of
Dialogue: 0,0:04:09.77,0:04:14.46,Default,,0000,0000,0000,,those two so c is going to be 13, 12, 1\Ntimes
Dialogue: 0,0:04:14.46,0:04:19.77,Default,,0000,0000,0000,,12 times 13, 12 times 12 is 144, plus\Nanother 12 is going to be 156.
Dialogue: 0,0:04:19.77,0:04:24.42,Default,,0000,0000,0000,,And just out of, just for fun you can\Nverify that
Dialogue: 0,0:04:24.42,0:04:27.24,Default,,0000,0000,0000,,this is going to be equal to 6 times their\Nsum.
Dialogue: 0,0:04:27.24,0:04:29.64,Default,,0000,0000,0000,,Their sum is, 26, 26 times 6 is 156
Dialogue: 0,0:04:29.64,0:04:34.50,Default,,0000,0000,0000,,so this one definitely worked, it\Ndefinitely worked for the
Dialogue: 0,0:04:34.50,0:04:36.79,Default,,0000,0000,0000,,constraints and it should because we\Nboiled down those constraints
Dialogue: 0,0:04:36.79,0:04:39.82,Default,,0000,0000,0000,,to a times b needed to be equal to 12.
Dialogue: 0,0:04:39.82,0:04:45.01,Default,,0000,0000,0000,,So let's try another one, 2 times 6, their\Nsum is
Dialogue: 0,0:04:45.01,0:04:50.40,Default,,0000,0000,0000,,8, and then if I were to take the product\Nof all
Dialogue: 0,0:04:50.40,0:04:55.33,Default,,0000,0000,0000,,of these, you get 2 times 6 is 12, times 8\Nis 96, 96.
Dialogue: 0,0:04:55.33,0:05:00.54,Default,,0000,0000,0000,,Then we could try 3 and 4, 3 plus 4 is 7,\N3
Dialogue: 0,0:05:00.54,0:05:06.33,Default,,0000,0000,0000,,times 4 is, 3 times 4 is 12 times 7,\Nactually I should
Dialogue: 0,0:05:06.33,0:05:11.29,Default,,0000,0000,0000,,have known, a times b is always 12 so we\Njust have to multiply 12 this last column.
Dialogue: 0,0:05:11.29,0:05:14.58,Default,,0000,0000,0000,,12 times 7 is 84, 12 times 7 is 84, and\Nthere
Dialogue: 0,0:05:17.11,0:05:21.15,Default,,0000,0000,0000,,aren't any others, you can't go, you\Ndefinitely can't go above 12, because then
Dialogue: 0,0:05:21.15,0:05:23.80,Default,,0000,0000,0000,,you'd have to deal with the non-integers,\Nyou'd have to deal with the fractions.
Dialogue: 0,0:05:23.80,0:05:25.79,Default,,0000,0000,0000,,You can't do the negative versions of\Nthese, because
Dialogue: 0,0:05:25.79,0:05:27.84,Default,,0000,0000,0000,,they all have to be positive integers, so\Nthat's
Dialogue: 0,0:05:27.84,0:05:30.73,Default,,0000,0000,0000,,it, those are all of the possible positive\Nintegers,
Dialogue: 0,0:05:30.73,0:05:33.01,Default,,0000,0000,0000,,we take their products, you get, you get\N12.
Dialogue: 0,0:05:33.01,0:05:35.11,Default,,0000,0000,0000,,You've essentially just factored 12.
Dialogue: 0,0:05:35.11,0:05:40.75,Default,,0000,0000,0000,,So, they want us, they want us to find the\Nsum of all possible values of N.
Dialogue: 0,0:05:40.75,0:05:43.91,Default,,0000,0000,0000,,Well these are all the possible values of\Nn.
Dialogue: 0,0:05:43.91,0:05:46.46,Default,,0000,0000,0000,,N is the product of those integers, so\Nlet's just take.
Dialogue: 0,0:05:46.46,0:05:51.50,Default,,0000,0000,0000,,Let's just take the sum, 6 plus 6 is 12\Nplus 4 is 16,
Dialogue: 0,0:05:51.50,0:05:56.51,Default,,0000,0000,0000,,1 plus 5 is 6 plus 9 is 15 plus 8 is 23,
Dialogue: 0,0:05:56.51,0:06:01.88,Default,,0000,0000,0000,,2 plus 1 is 3, so our
Dialogue: 0,0:06:01.88,0:06:07.19,Default,,0000,0000,0000,,answer is 336.