1
00:00:00,640 --> 00:00:04,700
I got this problem here
from the 2003 AIME exam.

2
00:00:04,700 --> 00:00:08,360
That stands for the American
Invitational Mathematics Exam.

3
00:00:08,360 --> 00:00:10,880
And this was actually the
first problem in the exam.

4
00:00:10,880 --> 00:00:17,230
The product N of three positive
integers is 6 times their sum,

5
00:00:17,230 --> 00:00:20,060
and one of the integers is
the sum of the other two.

6
00:00:20,060 --> 00:00:24,080
Find the sum of all
possible values of N.

7
00:00:24,080 --> 00:00:27,380
So we have to deal with
three positive integers.

8
00:00:27,380 --> 00:00:30,650
So we have three positive
integers right over here.

9
00:00:30,650 --> 00:00:33,080
So let's just think about
three positive integers.

10
00:00:33,080 --> 00:00:35,405
Let's call them a, b, and c.

11
00:00:35,405 --> 00:00:36,280
They're all positive.

12
00:00:36,280 --> 00:00:37,530
They're all integers.

13
00:00:37,530 --> 00:00:41,160
The product N of these
three positive integers--

14
00:00:41,160 --> 00:00:48,030
so a times b times c is equal
to N-- is 6 times their sum.

15
00:00:48,030 --> 00:00:51,110
This is equal to
6 times the sum.

16
00:00:51,110 --> 00:00:52,810
Let me do this in another color.

17
00:00:52,810 --> 00:00:54,450
So this is their product.

18
00:00:54,450 --> 00:00:57,690
So the product N of
three positive integers

19
00:00:57,690 --> 00:01:02,050
is 6 times their sum.

20
00:01:02,050 --> 00:01:05,310
So this is equal to 6
times the sum of those

21
00:01:05,310 --> 00:01:09,510
integers a plus b plus c.

22
00:01:09,510 --> 00:01:13,020
And one of the integers is
the sum of the other two.

23
00:01:19,950 --> 00:01:23,440
Well, let's just pick c
to be the sum of a and b.

24
00:01:23,440 --> 00:01:24,190
It doesn't matter.

25
00:01:24,190 --> 00:01:26,600
These are just names, and
we haven't said one of them

26
00:01:26,600 --> 00:01:28,330
is larger or less
than the other one.

27
00:01:28,330 --> 00:01:31,400
So let's just say
that a plus b is

28
00:01:31,400 --> 00:01:33,880
equal to c, that
one of the integers

29
00:01:33,880 --> 00:01:36,990
is the sum of the other two.
c is the sum of a plus b.

30
00:01:36,990 --> 00:01:41,990
Find the sum of all
possible values of N.

31
00:01:41,990 --> 00:01:45,160
So let's just try to do a
little bit of manipulation

32
00:01:45,160 --> 00:01:47,050
of the information we
have here, and maybe we

33
00:01:47,050 --> 00:01:50,146
can get some relationship
or some constraints

34
00:01:50,146 --> 00:01:51,770
on our numbers, and
then we can kind of

35
00:01:51,770 --> 00:01:54,050
go through all of
the possibilities.

36
00:01:54,050 --> 00:01:56,730
So let's see, we know that
a plus b is equal to c.

37
00:01:56,730 --> 00:02:02,360
So we can replace c
everywhere with a plus b.

38
00:02:02,360 --> 00:02:04,110
So this expression
right over here

39
00:02:04,110 --> 00:02:09,130
becomes ab, which is
just a times b, times c.

40
00:02:09,130 --> 00:02:11,780
But instead of c, I'm going to
write an a plus b over here.

41
00:02:15,500 --> 00:02:25,440
And then that is equal to
6 times a plus b plus c.

42
00:02:25,440 --> 00:02:31,360
And so once again, I'll replace
with the c with an a plus b,

43
00:02:31,360 --> 00:02:33,610
and then what does
this simplify to?

44
00:02:33,610 --> 00:02:36,150
So on the right-hand side,
we have 6 times a plus b

45
00:02:36,150 --> 00:02:37,020
plus a plus b.

46
00:02:37,020 --> 00:02:43,680
This is the same thing
as 6 times 2a plus 2b,

47
00:02:43,680 --> 00:02:45,520
just added the a's and the b's.

48
00:02:45,520 --> 00:02:46,700
And we can factor out a 2.

49
00:02:46,700 --> 00:02:49,740
This is the same thing as if
you take out a 2, 6 times 2

50
00:02:49,740 --> 00:02:53,320
is 12 times a plus b.

51
00:02:53,320 --> 00:02:55,856
The left-hand side
right over here

52
00:02:55,856 --> 00:03:01,720
is still a times b
or ab times a plus b.

53
00:03:01,720 --> 00:03:07,710
So ab times a plus b has got to
be equal to 12 times a plus b.

54
00:03:07,710 --> 00:03:09,430
So this is pretty
interesting here.

55
00:03:09,430 --> 00:03:12,610
We can divide both
sides by a plus b.

56
00:03:12,610 --> 00:03:17,140
We know that a plus b cannot be
equal to 0 since all of these

57
00:03:17,140 --> 00:03:20,940
numbers have to be
positive numbers.

58
00:03:20,940 --> 00:03:24,740
And the reason why I say that
is if it was 0, dividing by 0

59
00:03:24,740 --> 00:03:27,450
would give you an
undefined answer.

60
00:03:27,450 --> 00:03:30,130
So if we divide both
sides by a plus b,

61
00:03:30,130 --> 00:03:34,150
we get a times b is equal to 12.

62
00:03:34,150 --> 00:03:36,140
So all the constraints
that they gave us

63
00:03:36,140 --> 00:03:38,290
boiled down to this
right over here.

64
00:03:38,290 --> 00:03:41,530
The product of a and
b is equal to 12.

65
00:03:41,530 --> 00:03:43,726
And there's only
so many numbers, so

66
00:03:43,726 --> 00:03:46,100
many positive integers where
you if you take the product,

67
00:03:46,100 --> 00:03:46,950
you get 12.

68
00:03:46,950 --> 00:03:49,174
Let's try them out.

69
00:03:49,174 --> 00:03:50,590
So let me write
some columns here.

70
00:03:50,590 --> 00:03:54,290
Let's say a, b, c.

71
00:03:54,290 --> 00:04:00,070
And then we care
about their product,

72
00:04:00,070 --> 00:04:03,700
so I'll write that
over here, so abc.

73
00:04:03,700 --> 00:04:08,050
So if a is 1, b
is going to be 12.

74
00:04:08,050 --> 00:04:13,530
c is the sum of those two, so
c is going to be 13, 1 times

75
00:04:13,530 --> 00:04:15,380
12 times 13.

76
00:04:15,380 --> 00:04:21,959
12 times 12 is 144 plus
another 12 is going to be 156.

77
00:04:21,959 --> 00:04:24,620
And just for fun, you
can verify that this

78
00:04:24,620 --> 00:04:27,035
is going to be equal
to 6 times their sum.

79
00:04:27,035 --> 00:04:32,280
Their sum is 26,
26 times 6 is 156.

80
00:04:32,280 --> 00:04:33,530
So this one definitely worked.

81
00:04:33,530 --> 00:04:34,850
It definitely worked
for the constraints.

82
00:04:34,850 --> 00:04:37,100
And it should because we
boiled down those constraints

83
00:04:37,100 --> 00:04:40,060
to a times b need
to be equal to 12.

84
00:04:40,060 --> 00:04:41,720
So let's try another one.

85
00:04:41,720 --> 00:04:45,670
2 times 6, their sum is 8.

86
00:04:45,670 --> 00:04:48,200
And then if I were to take
the product of all of these,

87
00:04:48,200 --> 00:04:52,730
you get 2 times 6
is 12 times 8 is 96.

88
00:04:55,370 --> 00:04:58,830
And then we could try 3 and 4.

89
00:04:58,830 --> 00:05:01,154
3 plus 4 is 7.

90
00:05:01,154 --> 00:05:06,705
3 times 4 is 12 times 7.

91
00:05:06,705 --> 00:05:09,080
Actually, I should have known
the a times b is always 12,

92
00:05:09,080 --> 00:05:11,560
so you just have to multiply
12 times this last column.

93
00:05:11,560 --> 00:05:14,190
12 times 7 is 84.

94
00:05:17,080 --> 00:05:19,199
And there aren't any others.

95
00:05:19,199 --> 00:05:21,240
You definitely can't go
above 12 because then you

96
00:05:21,240 --> 00:05:22,580
would have to deal
with non-integers.

97
00:05:22,580 --> 00:05:23,840
You would have to
deal with fractions.

98
00:05:23,840 --> 00:05:25,520
You can't do the negative
versions of these

99
00:05:25,520 --> 00:05:27,436
because they all have
to be positive integers.

100
00:05:27,436 --> 00:05:28,120
So that's it.

101
00:05:28,120 --> 00:05:30,620
Those are all of the
possible positive integers

102
00:05:30,620 --> 00:05:33,070
where if you take their
products you get 12.

103
00:05:33,070 --> 00:05:35,300
We've essentially
just factored 12.

104
00:05:35,300 --> 00:05:41,620
So they want us to find the sum
of all possible values of N.

105
00:05:41,620 --> 00:05:43,420
Well, these are all
the possible values

106
00:05:43,420 --> 00:05:45,350
of N. N was the product
of those integers.

107
00:05:45,350 --> 00:05:47,920
So let's just take the sum.

108
00:05:47,920 --> 00:05:52,980
6 plus 6 is 12 plus
4 is 16, 1 plus 5

109
00:05:52,980 --> 00:06:01,740
is 6 plus 9 is 15 plus
8 is 23, 2 plus 1 is 3.

110
00:06:01,740 --> 00:06:04,750
So our answer is 336.