Counting Backwards in Steps of 6, 7, & 8

Everyone knows that it is not difficult to count forwards from 1 to any desired point. It is not difficult to do the same thing in steps of 2, 5, and 10. Counting forwards in steps other than these is more difficult but still relatively easy.

But what can we say about counting backwards? Some steps are easy but others; for example, 6, 7, or 8 are more difficult. If I were asked to do this, I might use a scheme that would make the chore easier. Let’s consider 7. If I subtract 7 from 10, the result will be 3. I can add 3 to a number, then subtract 10, I will find a number that is 7 less than that number. I’ve changed the problem to one that involves the simple operations of adding a number and subtracting 10. Let’s look at how this would work when counting downwards in steps of 7 from 100.

The first step would be computed from 100 + 3 -10. This would lead to 103 – 10 or 93.
The second step would be computed from 93 + 3 -10. This would lead to 96 – 10 or 86.
The third step would be computed from 86 + 3 -10. This would lead to 89 – 10 or 79.
The next steps would be as follows:

79 + 3 = 82 – 10 = 72
72 + 3 = 75 – 10 = 65
65 + 3 = 68 – 10 = 58
58 + 3 = 61 – 10 = 51
51 + 3 = 54 – 10 = 44
44 + 3 = 47 – 10 = 37
37 + 3 = 40 – 10 = 30
30 + 3 = 33 – 10 = 23
23 + 3 = 26 – 10 = 16
16 + 3 = 19 – 10 = 9
9 + 3 = 12 – 10 = 2

This final number can be checked for accuracy by doing the mathematical procedure called computing 100 modulo 7; that is dividing 100 by 7 and looking at only the remainder, which would be 2.  I can conclude that the series of number shown above are probably OK.

Let’s consider another example. Suppose we need to count from 100 downwards in steps of 6. We can compute the final number from computing 100 modulo 6. The remainder from dividing 100 by 6 is 4. The last number will be 4.

The number to add at each step is 10 – 6 or 4.

The numbers computed at all the required steps are these:

100 + 4 = 104 – 10 = 94
94 + 4 = 98 – 10 = 88
88 + 4 = 92 – 10 = 82
82 + 4 = 86 – 10 = 76
76 + 4 = 80 – 10 = 70
70 + 4 = 74 – 10 = 64
64 + 4 = 68 – 10 = 58
58 + 4 = 62 – 10 = 52
52 + 4 = 56 – 10 = 46
46 + 4 = 50 – 10 = 40
40 + 4 = 44 – 10 = 34
34 + 4 = 38 -10 = 28
28 + 4 = 32 – 10 = 22
22 + 4 = 26 – 10 = 16
16 + 4 = 20 – 10 = 10
10 + 4 = 14 = 10 = 4

A third example is where the step size is 8. The number to add at each step is 2. The final result number is to be 4. The results at the ends of all the steps are these:

100 + 2 = 102 – 10 = 92
92 + 2 = 94 – 10 = 84
84 + 2 = 86 – 10 = 76
76 + 2 = 78 – 10 = 68
68 + 2 = 70 – 10 = 60
60 + 2 = 62 – 10 = 52
52 + 2 = 54 – 10 = 44
44 + 2 = 46 – 10 = 36
36 + 2 = 38 – 10 = 28
28 + 2 = 30 – 10 = 20
20 + 2 = 22 – 10 = 12
12 + 2 = 14 – 10 = 4

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