LET A=8888 to the power 8888
B=sum of digits of A.
C=sum of digits of B
----------------------------
-------------and so on
unless; x=x
find the value of"x"
please do it fast as you can?
Answers
At prima facie it will be lil’ hard to guess that this problem can be solved very easily using the concept of Arithmetic Progression or by Remainder theorem.
First of all we need to know that when we are using Remainder theorem, we are indirectly using an AP. Confused?
Let us take an example to prove the fact. Let’s find out the remainder, when 352 is divided by 9. You can very easily find that the remainder will be 1, right? Now here’s the fun fact—write down an AP where the first term is 1 and the common difference is 9. Then you’ll find that the 40thterm of this series will be 352 and that’s our number. So basically in an AP, the remainder for all the terms remains the same when divided by their common difference. So keeping this concept in mind, let’s get back to our original problem.
In the problem, what they are basically asking for is—the first term of the AP of which 88888888is a part of. And as they have mentioned digit sum so we can use remainder theorem of 9 (which uses digit sum). Further we can use Fermat’s theorem to ease out the process of finding the remainder.
As per the Fermat’s theorem, the Euler number of 9 will be 6 and since 8888 and 9 are co-primes so
Rem[(8888^6)/9]=1 and by using this property we can very easily find out that Rem[(8888^8888)/9]=7
So the digit sum will come down to a single digit number i.e. 7.
Also to look it into another way, it is an arithmetic progression series where the first term is 7 and common difference is 9 and at some point of time 88888888will also appear in the series (don’t waste your time in finding out which term ). So the AP will look like
7,16, 25,………….88888888,…….
Answer:
If A=88888888 . B=sum of digits of A. C=Sum digits of B. D=Sum of digits of C and so on and so forth. In this series there will be a point where you’ll get sum of digit X=X itself. Find the value of X.
At prima facie it will be lil’ hard to guess that this problem can be solved very easily using the concept of Arithmetic Progression or by Remainder theorem.
First of all we need to know that when we are using Remainder theorem, we are indirectly using an AP. Confused?
Let us take an example to prove the fact. Let’s find out the remainder, when 352 is divided by 9. You can very easily find that the remainder will be 1, right? Now here’s the fun fact—write down an AP where the first term is 1 and the common difference is 9. Then you’ll find that the 40thterm of this series will be 352 and that’s our number. So basically in an AP, the remainder for all the terms remains the same when divided by their common difference. So keeping this concept in mind, let’s get back to our original problem.
In the problem, what they are basically asking for is—the first term of the AP of which 88888888is a part of. And as they have mentioned digit sum so we can use remainder theorem of 9 (which uses digit sum). Further we can use Fermat’s theorem to ease out the process of finding the remainder.
As per the Fermat’s theorem, the Euler number of 9 will be 6 and since 8888 and 9 are co-primes so
Rem[(8888^6)/9]=1 and by using this property we can very easily find out that Rem[(8888^8888)/9]=7
So the digit sum will come down to a single digit number i.e. 7.
Also to look it into another way, it is an arithmetic progression series where the first term is 7 and common difference is 9 and at some point of time 88888888will also appear in the series (don’t waste your time in finding out which term ). So the AP will look like
7,16, 25,………….88888888,…….
Step-by-step explanation:
Hope his answer will help you.