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

[Albert1]

$ 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

 

[kaliprasad]

$ 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

 

[Albert1]

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

 

[kaliprasad]

the answer is correct ,but X is not 4771 digits

Thanks
it is my mistake x is 4772 digits

 

[Albert1]

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)

 

[kaliprasad]

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

 

[Albert1]

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 )