Multiples of Nine

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.,

Weighted Digit Sum Proposition 1:
The Weighted Digit Sum of Any Multiple
of any Digit of Five or Greater is that Digit

Illustrations

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

WDS(16) = 2*1 + 6 = 8
WDS(24) = 2*2 + 4 = 8
WDS(32) = 2*3 + 2 = 8
WDS(40) = 2*4 + 0 = 8
WDS(48) = 2*4 + 8 = WDG(16) = 8
WDS(56) = 2*5 + 6 = WDS(16) = 8
WDS(64) = 2*6 + 4 = WDS(16) = 8
WDS(72) = 2*7 + 2 = WDS(16) = 8
WDS(80) = 2*8 + 0 = WDS(16) = 8
WDS(88) = 2*8 + 8 = WDS(24) = 8
WDS(96) = 2*9 + 6 = WDS(24) = 8

Here is an example of a three digit multiple of 8

WDS(14*8) = WDS(112) = WDS(WDS(11),2) = WDS(32) = 8

The weight for 7 is h=3. For the first few multiples of 7.

WDS(14) = 3*1 + 4 = 7
WDS( 21) = 3*2 + 1 = 7
WDS( 28) = 3*2 + 8 = WDS(14) = 7
WDS( 35) = 3*3 + 5 = WDS(14) = 7
WDS( 42) = 3*4 + 2 = WDS(14) = 7
WDS( 49) = 3*4 + 9 = WDS(21) = 7

Here are cases of 6 with its weight of 4.

WDS(12) = 1*4 + 2 = 6
WDS(18) = 1*4 + 8 = WDS(12) = 6
WDS(24) = 2*4 + 4 = WDS(12) = 6
WDS( 30) = 3*4 + 0 = WDS(12) = 6
WDS(36) = 3*4 + 6 = WDS(18) = 6
WDS(42) = 4*4 + 2 = WDS(18) = 6
WDS(48) = 4*4 + 8 = WDS(24) = 6
WDS(54) = 5*4 + 4 = WDS(24) = 6

Here are cases of 5 with its weight of 5.

WDS(10) = 1*5 + 0 = 5
WDS(15) = 1*5 + 5 = WDS(10) = 5
WDS(20) = 2*5 + 0 = WDS(10) = 5
WDS(25) = 2*5 + 5 = WDS(15) = 5
WDS(30) = 3*5 + 0 = WDS(15) = 5
WDS(35) = 3*5 + 5 = WDS(20) = 5
WDS(40) = 4*5 + 0 = WDS(20) = 5
WDS(45) = 4*5 + 5 = WDS(25) = 5

In contrast here are the results for m=4

WDS(8) = 0*6 + 8 = 8 ≠ 4
WDS(12) = 1*6 + 2 = 8 ≠ 4
WDS(16) = 1*6 + 6 = WDS(12) = 8 ≠ 4
WDS(20) = 2*6 + 0 = WDS(12) = 8 ≠ 4
WDS(24) = 2*6 + 4 = WDS(16) = WDS(12) = 8 ≠ 4
WDS(28) = 2*6 + 8 = WDS(20) = WDS(12) = 8 ≠ 4
WDS(32) = 3*6 + 2 = WDS(20) = 8 ≠ 4
WDS(36) = 3*6 + 6 = WDS(24) = WDS(16) = 8 ≠ 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.

WDS(6) = 0*7 + 6 = 6 ≠ 3
WDS(9) = 0*7 + 9 = 9 ≠ 3
WDS(12) = 1*7 + 2 = 9 ≠ 3
WDS(15) = 1*7 + 5 = WDS(12) = 9 ≠ 3
WDS(18) = 1*7 + 8 = WDS(15) = 9 ≠ 3
WDS(21) = 2*7 + 1 = WDS(15) = 9 ≠ 3
WDS(24) = 2*7 + 4 = WDS(18) = 9 ≠ 3
WDS(27) = 2*7 + 7 = WDS(21) = 9 ≠ 3

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.

Numbers Greater than Ten

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

The Weighted Digit Sum of any Multiple of a Number m>10 is either −m or Zero

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.)