Write down all Addition Sums to 11
Answers
Step-by-step explanation:
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Answer:
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Step-by-step explanation:
It is well known that the digits of multiples of nine sum to nine; i.e., 9→9, 18→1+8=9, 27→2+7=9, . . ., 99→9+9=18→1+8=9, 108→1+0+8=9, etc. Less well known is that the sum of digits of multiples of other numbers have simple patterns although not so simple as the case of nine. These are shown below:
Number Repeating Cycle
of Sum of Digits
of Multiples
2 {2,4,6,8,1,3,5,7,9}
3 {3,6,9,3,6,9,3,6,9}
4 {4,8,3,7,2,6,1,5,9}
5 {5,1,6,2,7,3,8,4,9}
6 {6,3,9,6,3,9,6,3,9}
7 {7,5,3,1,8,6,4,2,9}
8 {8,7,6,5,4,3,2,1,9}
9 {9,9,9,9,9,9,9,9,9}
10 {1,2,3,4,5,6,7,8,9}
11 {2,4,6,8,1,3,5,7,9}
12 {3,6,9,3,6,9,3,6,9}
13 {4,8,3,7,2,6,1,5,9}
It is asserted that the sum of digits follows a repeating sequence of length 9. This is so because for any decimal representation x, the number 10*x (ten times x) will have the same sum of digits. Multiplication by the base number 10 simply moves the digits one place to the left and puts a zero in the units place. More generally,
DigitSum(10k*n) = DigitSum(n).
This operation, multiplication by the k-th power of 10, moves the digits k places to the left and places k zeroes on the right. The sum of digits is the same.
Less obviously adding a nine at any place in a decimal representation reduces the digit by one and adds one to the digit in the next higher place, and thus the sum of the digits is not altered. That is to say,
DigitSum(n+9*10k) = DigitSum(n).
Going back to the above table, what is immediately suggested by the information presented there is that the sequence for a multiple digit number m is the sequence for the sum of the digits of m, the digit sum of m. For example, the sequence for 12 is the same as the sequence for 1+2=3. Likewise the sequence for 13 is the same as the sequence for 1+3=4. This of course also holds true for 10.
The sequences for 3 and 6 are composed of the subsequences {3,6,9} and {6,3,9}, respectively. Thus the sequences for these digits have three copies of the subsequences of length 3. The sequence for 9 is nine copies of a subsequence of length 1. Note that {3,6,9}=3*{1,2,3} and that {6,3,9}=3*{2,1,3}. It is also worth noting that for all digits except 3, 6 and 9 the length of the sequence is nine and that for 3, 6 and 9 the lengths of their sequences, 3 and 1, are factors of 9.
The structures of the length 9 sequences are noteworthy. Here is a naive perception of the structure of the sequences. For 2 the sequence is the even digits in ascending order then the odd digits also in ascending order. For 4 the sequence is two generally descending sequences interleaved; i.e., 4,3,2,1,9 with 8,7,6,5 interleaved. For 5 the sequences is also two interleaved sequences but ascending rather than descending; i.e., 5,6,7,8,9 with 1,2,3,4 interleaved. For 7 the sequence is a descending sequence of the odd digits starting with 7 then the even digits in descending order with 9 the odd digit left out coming last. For 8 the sequence is a single descending sequence of digits starting with 8 and finishing with the digit 9 which was left out of the descending sequence. Later a more mathematical explanation of the structure of the sequences will be given.