6
$\begingroup$

First I use the fact that if the sum of the digits of a number is divided by $9$ then the number is divided by $9$.

So : $88^0\equiv 1\ [9]$

$88^1\equiv 7\ [9]$

$88^2\equiv 4\ [9]$

$88^3\equiv 1\ [9]$

Then $88^{88}=\big(88^{29}\big)^3\times88^{1}\equiv 7 \ [9]$

I try to see the possibilities : we have $88^{88}\le 10^{176}$ which means that the sum of the digits is bounded by $9\times176=1584$. Then the sum of the sum is bounded by $1+45+81+36=163$. Then the sum of the sum of the sum is bounded by $1+54+27=82$. Then the sum of the sum of the sum of the sum is bounded by $8+9=17$.

So it could be $7$ or $16$ but $1+6=7$. That's why it's $7$ the final answer.

Thanks in advance !

$\endgroup$
  • $\begingroup$ To find the digital sum, you will have to compute $88^{88}$ explicitly. The reason being that taking remainder modulo $9$ does not at all simplify the calculation of the sum,it only provides a bound. $\endgroup$ – астон вілла олоф мэллбэрг Aug 31 '16 at 11:56
  • $\begingroup$ 88 doesn't look very compatible with the decimal system.. So I suspect there is no easy way round it. $\endgroup$ – Vim Aug 31 '16 at 11:57
  • 2
    $\begingroup$ Are you sure you don't mean the iterated digit sum? That's $7$, for the reason you mentioned. The answer to your question is $745$ by computer, but I don't see any short cut to seeing that. $\endgroup$ – lulu Aug 31 '16 at 11:58
  • $\begingroup$ @lulu by iterated you mean "the sum of the sum of the sum" ? $\endgroup$ – Maman Aug 31 '16 at 12:00
  • $\begingroup$ @Maman Indeed. This is the kind of sum (along with the mod-11 sum) that we use to check hand-written calculations. $\endgroup$ – Parcly Taxel Aug 31 '16 at 12:01
0
$\begingroup$

Eventually the iterated sum of the digits will be a single digit number. A number is congruent to the sum of its decimal digits mod $9$, so every iterated sum will be congruent to $7$ mod $9$ (assuming your calculation is correct). Therefore the final iterated sum of the digits is $7$.

$\endgroup$
  • $\begingroup$ Why do you exclude $16$ ? $\endgroup$ – Maman Aug 31 '16 at 12:22
  • $\begingroup$ @Maman The sum of the digits of $16$ is $7$. $\endgroup$ – Matt Samuel Aug 31 '16 at 12:22
  • $\begingroup$ because you should add the one and the six in 16 to get 7. $\endgroup$ – fleablood Aug 31 '16 at 12:23
  • $\begingroup$ Indeed no problem $\endgroup$ – Maman Aug 31 '16 at 12:24
  • $\begingroup$ ascertain using answer given by Dr Sonnhard Graubner $\endgroup$ – Piquito Aug 31 '16 at 12:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.