San José State University

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Thayer Watkins
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 Digit Sum Arithmetic

This material examines the arithmetic of digit sums. Some of the relationships are merely interesting, others are simply amazing. First some definitions are required.

The digit sum of a number, say 152, is just the sum of the digits, 1+5+2=8. If the sum of the digits is greater than nine then the process is repeated. For example, the sum of the digits for 786 is 7+8+6=21 and the sum of the digits for 21 is 3 so the digit sum of 786 is 3. In some places this concept of the digit sum of a number is called its reduced digit sum, but that terminology becomes too cumbersome. As in the preceding material the sum of the digit of a number may be different from its digit sum. The digit sum is the end result of repeatedly computing the sum of the digits until a single digit answer is obtained. The digit sum of a number n is denoted as DigitSum(n).

Here are illustrations of the properties of digit sum arithmetic. The first are pretty mundane but the latter ones can be justifiably characterized as amazing. In the following a and b stand for any terminating decimal number. This includes integers as a subcategory.

• The most famous property of digit sums is that the digit sum of all multiples of 9 is 9.

For examples, DigitSum(18)=9, DigitSum(27)=9, DigitSum(99)=9, etc.

• DigitSum(a+b) = DigitSum(DigitSum(a)+DigitSum(b))

Example, DigitSum(786+152) = DigitSum(938) = 2
DigitSum(786) + DigitSum(152) = DigitSum(3 + 8) = 2

• DigitSum(a-b) = DigitSum(DigitSum(a)-DigitSum(b))

Example, DigitSum(962-151) = DigitSum(811) = 1
DigitSum(962) - DigitSum(151) = DigitSum(8-7) = 1

This property has to be properly interpreted when the difference of the digit sums is negative or zero. It holds true in these cases as well but account must be taken of the fact that in digit sum arithmetic an answer of 9 is the same as an answer of 0. More on this later.

• DigitSum(a*b) = DigitSum(DigitSum(a)*DigitSum(b))

Example: DigitSum(35*16) = DigitSum(560) = 2
DigitSum(35)*DigitSum(16) = DigitSum(8*7) = DigitSum(56) = 2

• Division by single digit numbers other than multiples of 3 is equivalent to multiplication by a specific digit for that divisor. For example, division by 2 is equivalent to multiplication by 5. Thus DigitSum(32/2) = DigitSum(32*5) = DigitSum(160) = 7. which is the same as DigitSum(16). Division by 4 is equivalent to multiplication by 7 so DigitSum(20/4) = DigitSum(20*7) = DigitSum(140) = 5, which is correct.

The rule is DigitSum(a/b) = DigitSum(DigitSum(a)*Equivalent(DigitSum(b))) providing that DigitSum(b) is not a multiple of 3.

• DigitSum(Polynomial(a)) = DigitSum(Polynomial(DigitSum(a))

Example: Let Polynomial(a) = a²+a. Then Polynomial(11)=121+11=132 and thus DigitSum(Polynomial(11))=6. DigitSum(11)=2 so Polynomial(DigitSum(11))=4+2=6.

These properties are analyzed in the following readings: