The DigitSum Function

Let n it is in a number and also let Dec10(n) it is in the decimal representation of n.Let DigitSum(z) be the ultimate amount of digits of a decimal depiction z; i.e., if thesum of digits of a decimal depiction is better than nine then the amount of that number"sdigits is computed until a solitary digit is at some point obtained.This function, DigitSum( ), has several amazing properties; i.e.,DigitSum(x+y) = DigitSum(DigitSum(x) + DigitSum(y))DigitSum(x−y) = DigitSum(DigitSum(x) − DigitSum(y))DigitSum(x*y)=DigitSum(DigitSum(x)*y))DigitSum(x*y)=DigitSum(x*DigitSum(y))DigitSum(x*y)=DigitSum(DigitSum(x)*DigitSum(y))See digit Sums for an analysis and proof of these propositions.The proposition that DigitSum(x*y)=DigitSum(DigitSum(x)*y)) establishes that the sequences for themultiples that 12 and of 13 room the exact same as the sequences because that 3 (1+2) and also 4 (1+3), respectively.

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Decimal depiction of Numbers

In order for the following to make feeling one should stop thinking of a number in state ofits decimal representation and think that a number in regards to an appropriate variety of tallymarks so three would be (|||) and also eleven (||||||||||||).Consider just how one obtains the decimal depiction of a number. To get the last digit onedivides the number by ten and takes the remainder together the last digit. The last digit is subtracted from the number and the result divided through ten. Then the decimal representation ofthat quotient is sought. The procedure is repeated and the following the critical digit is obtained.An alternate characterization that the process of recognize the decimal depiction of a number is that the k-th power digit because that a whole number n is:ck = (trunc)%10where trunc<> method the fractional part is thrown away and also m%10 means the remainder afterdivision through 10. In terms of the terminology native the programming language Pascal theformula isck(n) = (n div 10k) mode 10The amount of the digits because that a number n is thenSum = Σk ck(n),but this is not necessarily the digit amount for the number. The process has to it is in repeatediteratively top top the sum of the digits.

The Explanation the the patterns of the Sequences

Consider two digits, a and b. If their sum is much less than ten thenDigitSum(a+b) = DigitSum(a)+DigitSum(b)and henceDigitSum(a+b) = DigitSum(DigitSum(a)+DigitSum(b)).But, if the amount of a and b is ten or much more then the decimal depiction of their amount is a one inthe ten"s place and (a+b−10) in the unit"s place. ThusDigitSum(a+b) = 1 + (a+b−10) = a+b−(10−1) = a+b−9For number the DigitSum(a)=a and DigitSum(b)=b for this reason DigitSum(DigitSum(a)+DigitSum(b)) = 1 + (a+b−10).Therefore for any type of two number a and also bDigitSum(a+b) = DigitSum(DigitSum(a)+DigitSum(b)).This uses as fine to the number in the k-th place. Thus the general proposition forany two decimal representations of numbers, x and yDigitSum(a+b) = DigitSum(DigitSum(a)+DigitSum(b)).For differences, if a and also b space digits and also a>b thenDigitSum(a−b) = DigitSum(DigitSum(a)−DigitSum(b)).On the other hand if aDigitSum(a−b) = DigitSum(DigitSum(a)−DigitSum(b)).Again this extends come the number in any kind of place in a decimal representation of a number.For any type of decimal number x and also y thenDigitSum(x±y) = DigitSum(DigitSum(x)±DigitSum(y)).Since multiplication is merely repeated enhancement it likewise follows thatDigitSum(x*y) = DigitSum(DigitSum(x)*y)DigitSum(x*y) = DigitSum(x*DigitSum(y))and finallyDigitSum(x*y) = DigitSum(DigitSum(x)*DigitSum(y)).It was previously detailed that for any kind of two number whose sum is better than ten,DigitSum(a+b) = 1 + (a+b−10) = a+b−(10−1) = a+b−9In basic then for any type of decimal representation xDigitSum(x) = Sumofdigits(x) − m*9where m is such the DigitSum(x) is diminished to a single digit.Another way of to express this is the the DigitSum because that a number n is just the remainder after division by 9; i.e., DigitSum(n)=(n%9). DigitSum arithmetic is simplyarithmetic modulo 9.For to compare the multiplication table for modulo 9 arithmetic is:Multiplication Table because that Modulo 9 ArithmeticIf the 0"s were replaced by 9"s and also the table rearranged therefore the first column i do not care the last column and the an initial rowbecomes the last heat the result would be identical to the table because that the sequences of number sums.
000000000
The Rearrangement that the Multiplication Table because that Modulo 9 ArithmeticWith 9 Substituted because that 0
123456789
246813579
369369369
483726159
516273849
639639639
753126429
876543219
999999999

Some Proofs

The proof of the building that the DigitSum of any type of multiple of ripe is equal to nine starts through the obvious propositionDigitSum(10*n) = DigitSum(n). From this it follows from the proposition that DigitSum(x−y)=DigitSum(x)−DigitSum(y) thatDigitSum((10−1)*n) = 0 and thusDigitSum(9*n) = 0but in modulo 9 arithmetic 0 is the very same as 9, soDigitSum(9*n) = 9The way to recognize the sequence because that 8 is that once 8 is included to any digit other than 0 or 1in any kind of place the number is diminished by 2 and one added to the number of the following place. Thisresults in a net decrease in the sum of digits of one. Thus when 8 is added to 8 the digitbecomes 6, a reduction of 2, and also 1 is included to the next greater digit, a net decrease in thesum of the digits of 1. Thus included 8 come 8 results in a sum of number of 7. Adding 8 come 7results in a amount of digits of 6, and so on under to including 8 come 1 which provides 9. Also this fits into the dominance in the feeling that if 1 were mitigate by 1 the an outcome would be 0 i m sorry is indistinguishable to 9 modulo 9. Likewise adding 7 come a digit reduces it by 3 and adds 1 to thedigit in the following place, a net reduction in the amount of number of 2. Therefore when 7 is added to 7the sum of number is decreased to 5. Once 7 is added to 5 the amount of number is diminished to 3.When 7 is included to 3 the sum of digits is decreased to 1. If 7 is added to 1 the result is 8,but the 8 can be take into consideration as a reduction of 1 by 2 modulo 9. The enhancement of 7 to 8 resultsin a sum of number of 6 and so on under to a amount of number of 2. The addition of 7 come 2results in a sum of number of 9, however that 9 deserve to be considered 0 modulo 9 and thus it is a reduction of 2.

Generalization to other Number Bases

There is naught special about 9; that is simply the number base ten less one. The digitsum sequences because that multiples in thehexadecimal (base 16) number device is:
NumberRepeating Cycleof amount of Digitsof Multiples
22,4,6,8,a,c,e,1,3,5,7,9,b,d,f
33,6,9,c,f,3,6,9,c,f,3,6,9,c,f
44,8,c,1,5,9,d,2,6,a,e,3,7,b,f
55,a,f,5,a,f,5,a,f,5,a,f,5,a,f
66,c,3,9,f,6,c,3,9,f,6,c,3,9,f
77,e,6,d,5,c,4,b,3,a,2,9,1,8,f
88,1,9,2,a,3,b,4,c,5,d,6,e,7,f
99,3,c,6,f,9,3,c,6,f,9,3,c,6,f
aa,5,f,a,5,f,a,5,f,a,5,f,a,5,f
bb,7,3,e,a,6,2,d,9,5,1,c,8,4,f
cc,9,6,3,f,c,9,6,3,f,c,9,6,3,f
dd,b,9,7,5,3,1,e,c,a,8,6,4,2,f
ee,d,c,b,a,9,8,7,6,5,4,3,2,1,f
ff,f,f,f,f,f,f,f,f,f,f,f,f,f,f,f
In general, because that numbers of base b the digit sums are equivalent to arithmetic modulo (b−1). The sequences which contain subsequences room the multiples of the determinants of (b−1)other 보다 unity.In the instance of b=ten these components are 3 and also 9. Because of this there are subsequences for 3,6,9.

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The lengths the the subsequences are (b−1) split by the factors; i.e.,in the case of b=ten 3 and 1. For b=sixteen the factors are three, fiveand fifteen and also so subsequences happen for 3,6,9,c,f,5,a and the lengths that thesubsequences room 3, 5 and also 1.HOME page OF applet-magicHOME page OF Thayer Watkins