1 00:00:00,730 --> 00:00:04,700 I got this problem here from the 2003 AIME exam. 2 00:00:04,700 --> 00:00:08,340 That stands for the American Invitational Mathematics Exam, and 3 00:00:08,340 --> 00:00:10,318 this was actually the first problem in the exam. 4 00:00:10,318 --> 00:00:17,330 The product N of three positive integers is six times their sum, 5 00:00:17,330 --> 00:00:20,180 and one of the integers is the sum of the other two. 6 00:00:20,180 --> 00:00:24,160 Find the sum of all possible values of N. 7 00:00:24,160 --> 00:00:27,400 So we have to deal with three positive integers. 8 00:00:27,400 --> 00:00:30,420 So we have three positive integers right over 9 00:00:30,420 --> 00:00:32,910 here, so let's just think about three positive integers. 10 00:00:32,910 --> 00:00:35,450 Let's call them a, b and c. 11 00:00:35,450 --> 00:00:37,670 They're all positive, they're all integers. 12 00:00:37,670 --> 00:00:41,110 The product N, of these 3 positive, these 3 positive integers. 13 00:00:41,110 --> 00:00:48,120 So a times b times c is equal to N, is 6 times their sum. 14 00:00:48,120 --> 00:00:50,660 This is equal to 6 times the sum. 15 00:00:50,660 --> 00:00:52,840 Let me do this in another color. 16 00:00:52,840 --> 00:00:54,430 So, this is their product. 17 00:00:54,430 --> 00:01:02,080 So, the product N of three positive integers is 6 times, is 6 times their sum. 18 00:01:02,080 --> 00:01:09,650 So, this is equal to six times the sum of those integers, a plus b plus c. 19 00:01:09,650 --> 00:01:12,980 And one of the integers is the sum of the other two. 20 00:01:14,170 --> 00:01:19,240 One, one of the integers is the sum of the other two. 21 00:01:19,240 --> 00:01:22,660 Well let's just pick c to be the sum of a and b. 22 00:01:22,660 --> 00:01:25,600 We could do, it doesn't matter, these are just names and we 23 00:01:25,600 --> 00:01:28,270 haven't said one of them is larger or less than the other one. 24 00:01:28,270 --> 00:01:32,220 So let's just said that a plus b is equal to c. 25 00:01:32,220 --> 00:01:37,090 The one of the integers is the sum of the other two, c is the sum of a plus b. 26 00:01:37,090 --> 00:01:40,913 Find the sum of all possible values of N. 27 00:01:40,913 --> 00:01:44,420 So let's just try to do a little bit of 28 00:01:44,420 --> 00:01:47,263 manipulation of the information of what we have here and maybe, 29 00:01:47,263 --> 00:01:50,570 we can get some relationship or some constraints on our 30 00:01:50,570 --> 00:01:54,050 numbers and then we can kinda go through all the possibilities. 31 00:01:54,050 --> 00:01:56,730 So let's see, we know that a plus b is equal to c. 32 00:01:56,730 --> 00:02:02,970 So we could replace c, we can replace c everywhere with a plus b, so this 33 00:02:02,970 --> 00:02:09,220 expression right over here becomes ab, which is just a times b, times c, but 34 00:02:09,220 --> 00:02:14,360 instead of c, I'm gonna write an a plus b over here, a plus b, 35 00:02:15,620 --> 00:02:22,140 and then that is equal to 6 times, that is equal to 6 times a plus b, 36 00:02:22,140 --> 00:02:24,958 a plus b plus c. 37 00:02:24,958 --> 00:02:30,790 And so, once again I'll replace the c with an a plus b. 38 00:02:30,790 --> 00:02:33,710 And then what does this simplify to. 39 00:02:33,710 --> 00:02:37,020 So on the right hand side, we have 6 time a plus b plus a plus b. 40 00:02:37,020 --> 00:02:43,690 This is the same thing as 6 times 2a plus 2b, 2a plus 2b, 41 00:02:43,690 --> 00:02:46,820 just added the a's and the b's and we can factor out a 2. 42 00:02:46,820 --> 00:02:52,350 This is the same thing as if you take out a 2, 6 times 2 is 12 times a 43 00:02:52,350 --> 00:02:57,440 plus b, the left hand side over here is still, is still a 44 00:02:57,440 --> 00:03:02,434 times b, or a b, times a plus b, so ab times 45 00:03:02,434 --> 00:03:07,810 a plus b has got to be equal to 12 times a plus b. 46 00:03:07,810 --> 00:03:12,680 So this is pretty interesting here, we can divide both sides by a plus b. 47 00:03:12,680 --> 00:03:15,740 We know that a plus b won't be equal to, cannot be 48 00:03:15,740 --> 00:03:19,480 equal to zero since all of these numbers have to be positive numbers. 49 00:03:19,480 --> 00:03:22,060 So if we divide both sides, and the reason why I say that is you, 50 00:03:22,060 --> 00:03:27,450 if you divide, if it was zero, dividing by zero would give you an undefined answer. 51 00:03:27,450 --> 00:03:34,130 So if we divide both sides by a plus b, we get a times b is equal to twelve. 52 00:03:34,130 --> 00:03:37,430 So all the constraints that they gave us boiled down to 53 00:03:37,430 --> 00:03:40,590 this right over here, the product of a and b is 54 00:03:40,590 --> 00:03:43,990 equal to 12 and there's only so many numbers, so many 55 00:03:43,990 --> 00:03:46,900 positive integers where, if you take their product, you get twelve. 56 00:03:46,900 --> 00:03:48,520 Let's try them out. 57 00:03:48,520 --> 00:03:49,520 Let's try them out. 58 00:03:49,520 --> 00:03:50,620 So let me try some columns here. 59 00:03:50,620 --> 00:03:58,730 Let's say a, b, c, and then we care, we care about their product. 60 00:03:58,730 --> 00:03:59,980 We care about their product. 61 00:03:59,980 --> 00:04:01,106 So I'll write that over here. 62 00:04:01,106 --> 00:04:03,800 So a, b, c. 63 00:04:03,800 --> 00:04:09,770 So if a is 1, if a is 1, b is going to be 12, c is the sum of 64 00:04:09,770 --> 00:04:14,460 those two so c is going to be 13, 12, 1 times 65 00:04:14,460 --> 00:04:19,769 12 times 13, 12 times 12 is 144, plus another 12 is going to be 156. 66 00:04:19,769 --> 00:04:24,420 And just out of, just for fun you can verify that 67 00:04:24,420 --> 00:04:27,240 this is going to be equal to 6 times their sum. 68 00:04:27,240 --> 00:04:29,640 Their sum is, 26, 26 times 6 is 156 69 00:04:29,640 --> 00:04:34,500 so this one definitely worked, it definitely worked for the 70 00:04:34,500 --> 00:04:36,788 constraints and it should because we boiled down those constraints 71 00:04:36,788 --> 00:04:39,820 to a times b needed to be equal to 12. 72 00:04:39,820 --> 00:04:45,007 So let's try another one, 2 times 6, their sum is 73 00:04:45,007 --> 00:04:50,400 8, and then if I were to take the product of all 74 00:04:50,400 --> 00:04:55,330 of these, you get 2 times 6 is 12, times 8 is 96, 96. 75 00:04:55,330 --> 00:05:00,540 Then we could try 3 and 4, 3 plus 4 is 7, 3 76 00:05:00,540 --> 00:05:06,330 times 4 is, 3 times 4 is 12 times 7, actually I should 77 00:05:06,330 --> 00:05:11,290 have known, a times b is always 12 so we just have to multiply 12 this last column. 78 00:05:11,290 --> 00:05:14,580 12 times 7 is 84, 12 times 7 is 84, and there 79 00:05:17,110 --> 00:05:21,150 aren't any others, you can't go, you definitely can't go above 12, because then 80 00:05:21,150 --> 00:05:23,800 you'd have to deal with the non-integers, you'd have to deal with the fractions. 81 00:05:23,800 --> 00:05:25,790 You can't do the negative versions of these, because 82 00:05:25,790 --> 00:05:27,840 they all have to be positive integers, so that's 83 00:05:27,840 --> 00:05:30,730 it, those are all of the possible positive integers, 84 00:05:30,730 --> 00:05:33,010 we take their products, you get, you get 12. 85 00:05:33,010 --> 00:05:35,110 You've essentially just factored 12. 86 00:05:35,110 --> 00:05:40,750 So, they want us, they want us to find the sum of all possible values of N. 87 00:05:40,750 --> 00:05:43,910 Well these are all the possible values of n. 88 00:05:43,910 --> 00:05:46,460 N is the product of those integers, so let's just take. 89 00:05:46,460 --> 00:05:51,500 Let's just take the sum, 6 plus 6 is 12 plus 4 is 16, 90 00:05:51,500 --> 00:05:56,510 1 plus 5 is 6 plus 9 is 15 plus 8 is 23, 91 00:05:56,510 --> 00:06:01,880 2 plus 1 is 3, so our 92 00:06:01,880 --> 00:06:07,189 answer is 336.