Dependent on m m ?

Number Theory Level 3

For each number n n , we define the "Digital Sum" as sum of the digits of n n and repeat the process until the result is a single digit number.

Example :

For 1234 → 1 + 2 + 3 + 4 → 10 → 1 + 0 = 1 1234 \rightarrow 1 + 2 + 3 + 4 \rightarrow 10 \rightarrow 1 + 0 = 1

Find the digital sum of 4617 … 4617 ⏟ m times \underset { m \text{ times} }{ \underbrace { 4617\ldots 4617} } .

m > 0 m > 0 .

Inspired by 2016 Digital Sum


The answer is 9.

A Former Brilliant Member by A Former Brilliant Member

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2 solutions

The digital sum for 4617 is 4 + 6 + 1 + 7 = 18 = 1 + 8 = 9. Nine is a special number where 9, n times or m times the digital number remains the same. For Example : 9 × 123 = 1107 , Digital Sum is 9. This same in all the cases with the number nine. So the Digital sum is always nine for any number of terms. \large \displaystyle \text{The digital sum for } 4617 \text{ is } 4+6+1+7 = 18 = 1+ 8 = 9.\\ \large \displaystyle \text{ Nine is a special number where 9, n times or m times the digital number remains the same.}\\ \large \displaystyle \text{For Example : } 9 \times 123 = 1107, \text{ Digital Sum is } 9.\\ \large \displaystyle \text{This same in all the cases with the number nine.}\\ \large \displaystyle \text{So the Digital sum is always nine for any number of terms.}

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Saakshi Singh
Apr 11, 2016

4617 repeated m times , then digital sum is 18m. Broken again into 10m + 8m or m + 8m = 9m. Now whatever the value of m , digit sum will be nine.

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