A number has 2016 digits of 4, followed by 2015 digits of 8 and one digit of 9. Find the sum of the digits of square root of this number.

Question

I can't solve the problem describing below:

A number has 2016 digits

of 4, followed by 2015 digits of 8 and
one digit of 9 like 4444.....8888.....9. Find the sum of the digits
of square root of this number.
This problem seems very very tough to me.Can anyone help me please?

Answer 1

Hi,

The first thing  we have to figure out is that is this number Perfect Square or not?! Probably, yes.

That is, there is a number m such that m^2=44...48...89.

 

Most probably this number is a Perfect Square because

a) it  ends with 9 (89 is not Perfect  square but ...89 can be);

b) its ten's digit is even =4;

c) digital root is equal 1: 2016*4+2015*8+9=24193, 2+4+1+9+3=19, 1+9=10 and finally  1+0=1.

 

Now how to find that m.

 

We can do the following: 4...48....89=4...48....88+1, So

 

4...48....88+1=4[10⁴⁰³¹+10⁴⁰³⁰+....+10^2016] + 8[10²⁰¹⁵+10²⁰¹⁴+....+10+1]+1=

=4*10²⁰¹⁶[10²⁰¹⁵+....+10+1] + 8[10²⁰¹⁵+10²⁰¹⁴+....+10+1]+1=

|Here we have finite geometric series (progression)|=

=[4*10²⁰¹⁶+8][10²⁰¹⁵+....+10+1]+1=

+[(10²⁰¹⁶  -1)/(10-1)]*[4*10²⁰¹⁶+8]+1=(4/9)(10²⁰¹⁶ -1)*(10²⁰¹⁶ +2)+1=

=(4/9)((10²⁰¹⁶)² +10²⁰¹⁶ -2)+1=

= (4/9)[(10²⁰¹⁶+1/2)²-2-1/4]+1=

=(4/9)*[(10²⁰¹⁶+1/2)²-9/4]+1=

=(4/9)*(10²⁰¹⁶+1/2)²-1+1=

=(4/9)(10²⁰¹⁶+1/2)^2.

 

So what we obtained is

4...48...89=(4/9)(10²⁰¹⁶+1/2)²! which a PERFECT SQUARE, that is

 

SQRT[4...48...89]=(2/3)(10²⁰¹⁶+1/2)

 

Now let's look at (2/3)(10²⁰¹⁶+1/2)!

 

(2/3)(10²⁰¹⁶+1/2)=(1/3)(2*10²⁰¹⁶+1)=(1/3)(2*100....0+1)=(1/3)*20....01. (here we have 2016 times 0's!)

 

Sum of digits of the number 20...01 is equal to 3, hence the it divides by 3.

 

20....01 consists of 2, 2015 zeros and 1. Then 20...01/3=66...67 (2014 6's and then 7. Now, sum up 6+6+....+6+7=6*2014+7=12091. This is the correct answer. (What I said before 1 was wrong (Sorry)). That was just to check that 20...01 divides by 3! Thus, eventually, the answer you were looking for is 12091.

I hope I helped somehow.

 

I know there is other ways to explain, too.

Please feel free to ask any questions, make comments. 

If it is not clear can provide a scan copy of my handwriting.