Download 35DigitSums - Learning Wrexham

Digit Sums of the factors
of a number
An Investigation
The idea:
Take a whole number say 22
Write down its factors.... 1, 2, 11 and 22
Add the digits of these factors together
 1+2+1+1+2+2 = 9
Repeat the process with this new number i.e.
factors are 1, 3 and 9
Adding these digits again gives  1+3+9 = 13
Repeat the process i.e. factors are 1, 13
Adding these digits again gives  1+1+3 = 5
Repeat the process i.e. factors are 1 and 5
Adding these digits again gives  1+5 = 6
Repeat the process i.e. factors are 1, 2, 3 and 6
Adding these digits again gives  1+2+3+6 = 12
Repeat the process i.e. factors are 1, 2, 3, 4, 6
and 12
Adding these digits again
gives  1+2+3+4+6+1+2 = 19
Repeat the process i.e. factors are 1 and 19
Adding these digits again gives  1+1+9 = 11
Repeat the process i.e. factors are 1 and 11
Adding these digits again gives  1+1+1 = 3
Repeat the process i.e. factors are 1 and 3
Adding these digits again gives  1+3 = 4
Repeat the process i.e. factors are 1, 2 and 4
Adding these digits again gives  1+2+4 = 7
Repeat the process i.e. factors are 1 and 7
Adding these digits again gives  1+7 = 8
Repeat the process i.e. factors are 1, 2, 4, 8
Adding these digits again gives  1+2+4+8 = 15
Repeat the process i.e. factors are 1, 3, 5, 15
Adding these digits again gives  1+3+5+1+5 =
15
----- this repeats again and again ------The sequence of digit sums obtained is therefore
22  9  13  5  6  12  19  11
 3  4  7  8  15  15 - - The digit sums repeat at 15 from now on.
Repeat the procedure above for yourself with
other numbers and see if you can answer the
question below. To speed up your work you
should note that if for example you started with
the number 12 or say 19 then you would also
end up at 15 as these are part of the sequence
above.
Question:
If you start with any positive whole numbers
(other than 1 ) do you always end up at 15?
Record your results in a table or diagram.
Solution
40
32
63
16
10
58 51
17
9
66
68
75
All numbers shown
15
31
5
76
41
28
36
34
71
53
13
25
46
18
30
27
22
70
55
29
38
23
45
6
33
12
42
47 19
73
24
11
2
37
77
69
72
60
64
74
3
4
65
35
21
7
50
49
59
44
8
14 67
43
61
15
62
20
39
52
26
52
57
54
48
56