2. Digit Sums of the factors of a number An Investigation

3. 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

4. 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

5. 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.

6. 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.

7. Solution