Brain teasing problem in number theory

[sagardip]

Homework Statement

Let X=10000000099 represent an eleven digit no. and let Y be a four digit no. which divides X.Find the sum of the digits of the four digit no. Y.

Homework Equations

None

The Attempt at a Solution

I guess I have to factorize X. But it is really difficult to do so as i can't find a prime which divides it. Moreover i am in search for a systematic solution and not any trial method.Any help in this regard would be sincerely appreciated. Thank You.

 

[Dickfore]

This is the prime factor expansion of the given number:

$$X = 10,000,000,099 = 23 \times 9901 \times 43913$$

(see http://www.wolframalpha.com/input/?i=FactorInteger[10000000099 for reference)

What are the possible four digit divisors of X?

 

[sagardip]

How did you find it out

 

[awkward]

Here is an approach which is potentially less computation-intensive.

If we call the 11-digit number N, it's easy to see that
N = 10^10 + 10^2 -1, so N = P(10), where
P(x) = x^10 + x^2 - 1.

If you were a genius, I guess it would be obvious that
P(x) = (x^4 - x^2 + 1) (x^6 + x^4 - 1).

Confession: Not being a genius, I used a computer algebra system.
I guess you can drive this identity from x^5 + x - 1 = (x^2 - x + 1) (x^3 + x^2 - 1), but that's not obvious to me either.

If you now let x = 10, we have
N = (10^4 - 10^2 + 1) (10^6 + 10^4 - 1),
and 10^4 - 10^2 + 1 = 9901.

This isn't a complete solution, because it's not clear from the above that 9901 is the only 4-digit divisor.

 

[sagardip]

Thanks
I understood it well and it is a very good solution