What is the Sum of Digits in the Power of Three Raised to 10,000?

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Discussion Overview

The discussion revolves around calculating the sum of the digits of the number \(3^{10000}\) and subsequently the sums of those sums, denoted as \(A\), \(B\), and \(C\). Participants explore the implications of the number of digits in \(X\) and how it affects the ranges of \(A\), \(B\), and \(C\).

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants assert that \(X = 3^{10000}\) has approximately 4771 digits, leading to the conclusion that \(A < 4771 \times 9\) or \(A < 50000\).
  • Others challenge the digit count, stating that \(X\) actually has 4772 digits, which alters the bounds for \(A\).
  • It is proposed that \(B < 41\) and thus \(C \leq 12\) if \(B\) were 39.
  • Some participants suggest that since \(X \mod 9 = C \mod 9\), \(C\) can only be 0 or 9, concluding that \(C\) cannot be zero, hence \(C = 9\).
  • There is a correction regarding the ranges of \(A\), \(B\), and \(C\), with some participants indicating that the initial ranges provided were incorrect.
  • One participant suggests a more rigid approach to the ranges, stating \(A\) is a 5-digit number and \(B < 5 \times 9 = 45\), leading to \(C < 2 \times 9 = 18\).

Areas of Agreement / Disagreement

Participants generally disagree on the number of digits in \(X\) and the corresponding ranges for \(A\), \(B\), and \(C\). While some calculations and conclusions are accepted by multiple participants, the overall discussion remains unresolved regarding the exact values and ranges.

Contextual Notes

There are limitations in the assumptions regarding the number of digits in \(X\) and the subsequent calculations for \(A\), \(B\), and \(C\). The discussion reflects uncertainty in these values and their implications.

Albert1
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$ X=3^{10000}$
Given $log_{10} 3=0.4771$
All the digits sum of X =A
All the digits sum of A =B
All the digits sum of B =C
please find C
 
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Albert said:
$ X=3^{10000}$
Given $log_{10} 3=0.4771$
All the digits sum of X =A
All the digits sum of A =B
All the digits sum of B =C
please find C

this is 4771 digits so A < 4771 * 9 or A < 50000

B < 41 so C <= 12 ( if b were 39)

X mod 9 = C mod 9

ax x mod 9 = 0 so C= 0 or 9 as it cannot be 18

as C cannot be zero it is 9
 
kaliprasad said:
this is 4771 digits so A < 4771 * 9 or A < 50000

B < 41 so C <= 12 ( if b were 39)

X mod 9 = C mod 9

ax x mod 9 = 0 so C= 0 or 9 as it cannot be 18

as C cannot be zero it is 9

the answer is correct ,but X is not 4771 digits
 
Albert said:
the answer is correct ,but X is not 4771 digits

Thanks
it is my mistake x is 4772 digits
 
A < 4771 * 9 or A < 50000

B < 41 so C <= 12

here the range of A,B,and C should be edited

(the range of A,B,and C are not correct)
 
Albert said:
A < 4771 * 9 or A < 50000

B < 41 so C <= 12

here the range of A,B,and C should be edited

(the range of A,B,and C are not correct)

so I replace it with correct solution

A < 4772 * 9 or < 47720 or < 50000

I have taken for A to be the worst case. hence A and B range is OK
 
A is 5 digits number
$B<5\times 9=45$
B is 2 digits number
$C<2\times 9=18$
I think your range is OK (more rigid )
 
Last edited:

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