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of Multiples of Any Digit of Five or Greater is that Digit |
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It is well known that the digit sums of all multiples of 9 are 9. Digit sum means that the sum of the digits of any result is computed until the result is a single digit. For example, consider 11*9=99. The sum of the digits of 99 is 18 and the sum of the digits of 18 is 9. Thus the digit sum of 99 is 9.
A far more general relationship prevails; i.e.,
Define the weight h for a digit m as 10−m. The weighted sum of two digits ab of a number is h*a+b. For example, the weight for 8 is 2. Consider the two digit multiples of 8; 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, and 96. Here are their weighted digit sums WDS
Here is an example of a three digit multiple of 8
The weight for 7 is h=3. For the first few multiples of 7.
Here are cases of 6 with its weight of 4.
Here are cases of 5 with its weight of 5.
In contrast here are the results for m=4
While the weighted digit sums of multiples of 4 are not equal to 4 they are equal a value equivalent to 4 in terms their remainders upon division by 4.
Now consider the weighted digit sums of multiples of 3. For m=3, h=7.
As with the case of m=4 the weighted sums of multiples of 3 are multiples of 3. Thus while the proposition under consideration does not hold for m<5 some more general proposition would hold. \
The proof of this particular proposition is given Elsewhere and a more general proposition is dealt with in Weighted Digit Sums as Remainders.
Consider m=11. Its weight h is equal to 10−11= −1. Example:
WDS(12*11) = WDS(132) = (−1)1 + 3 + (−1)2 = 0
Now consider m-12. Its weight h=10−12 = −2. Examples:
WDS(4*12) = WDS(48) = (−2)*4 + 8 = 0
WDS(12*12) = WDS(144) = WDS(WDS(14),4) = WDS(24) = (−2)*2 + 4 = 0
A Tentative General Rule
A wild example: Let m=19. Then h=−9. Consider 38=2*19. WDS(38)=(−9)*3 + 8 = −27 +8 = −19.
But the general rule is not fully formulated. Take m=20 so h=−10. Consider 40=2*20. WDS(40)=(−10)*4 = −40. WDS(−40)=(−10)(−4)=40.
(To be continued.)
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