August's Fatherly Digit Sum

Number Theory Level 4

Let $S(n)$ denote the digit sum of a positive integer $n$ (sum of all its digits base 10). Then, let $A, B, C$ be the non-negative integers satisfying the system of equations below. $\begin{aligned} S\left(5^A\right) + S\left(5^B\right) & = S\left(5^C\right)\\ A + B & = C \end{aligned}$ Determine the minimum value of $(A + B + C)$ .

The answer is 8.

2 solutions

Niranjan Khanderia
Aug 11, 2018

$S(5^1=5)=5\\ S(5^2=25)=7\\ S(5^3=125)=8\\ S(5^4=625)=13\\ ~~~~\\ S(5^1)~~ +~~S(5^3)=S(5^4)\\ \therefore~~A +B+C=1+3+4=\Huge~~\color{#D61F06}{8}.$

X X
Aug 9, 2018

Just start from $C=2$ and place in $A$ and $B$ to check. Checked to $C=4$ , and discover $A=1,B=3$ fits!

$1+3=4,\space5+(1+2+5)=6+2+5$

So the minimum is $1+3+4=8$ . (Also, this problem was posted on 8/8)

Yes, and the date was August 8, 2018 (though I posted days later). Happy Lucky 8!

Michael Huang - 2 years, 10 months ago

@X X , unlike Chinese, except for hyphen (-), we need a space after a punctuation mark such as comma (,), period or full-stop (.), exclamation mark (!), colon (:), semi-colon (;) and others. It will be marked as grammatically wrong in Microsoft Word.

Chew-Seong Cheong - 2 years, 10 months ago

Thanks for the information. I will keep this in mind.

X X - 2 years, 10 months ago

Fun fact: As of 3:09 PM (EST), August 11, 2018, we have 22 solvers out of 25 attempts. That surely makes about 88% correct as shown in Brilliant! What a coincidence! Another 8! XD

Michael Huang - 2 years, 10 months ago

This problem is about the power of 5, and I am curious about if it changed to the power of 8.

X X - 2 years, 10 months ago

Yep, but I didn't even think about that one. Though, it is know that the powers of $8$ follows the pattern of $2$ . Thus, I decide to consider only the prime numbers. :)

Michael Huang - 2 years, 10 months ago

@Michael Huang – I discovered that the digit sum of power of 8 follows a strange rule: There are many solutions of $S(8^n)=S(8^{n+2})$ , but not always.

This property is the reason I'm curious about if there is an answer if the problem changed to the power of 8.

X X - 2 years, 10 months ago

@X X – Yes, but that hasn't been proven yet. However, I can look into that when I can.

Michael Huang - 2 years, 10 months ago

@Michael Huang – You can view this

X X - 2 years, 10 months ago

@X X – Yes, I see that you found the sequence, relating to the case. I also see that for some integers $n$ , $S(8^n) = S(8^{n + 2})$ . I notice that some $n$ makes $S(S(8^n)) = 8$ or similar. Yet, the proof is difficult to find without the generalized sum formula.

The reason why I posted this problem is to find the way to get the result $8$ , which is for the special Chinese holiday.

Michael Huang - 2 years, 10 months ago

I wonder if there is a solution without trial and error!

Atomsky Jahid - 2 years, 10 months ago