Question

Let m = 444...
444 and n = 999...999 (both have 2010 digits). What is the 2010th digit of the product mn?

How to solve this kind of problem?

Answer 1

What if you decide to (n+1)*m?

Answer 2

That would make (n+1)*m end in a lot of zeros. But you have 1 too much of m. So you subtract m.
I'm just thinking aloud, this could be completely wrong thinking :D

Answer 3

I'm going with a 3.

Answer 4

I tried plugging 44444*99999 into my calculator and then counted the number on position 5.
Then I added to numbers 444 444*999 999, and counted the number on position 6 and so on. Everytime, the number was 3.

Answer 5

My guess is, the n:th number is going to be a 3 if you have n digits in both factors

Answer 6

Don't take my words for granted though, could be completely wrong. After all, I got no general proof. :D

Answer 7

the answer is correct .. but i cannot understand your solution ..

Answer 8

That's the thing, it's empirical. :D

Answer 9

No, it isn't the logic in the second comment is right...

Answer 10

I see!

Answer 11

If you multiply m*(n+1) you get 444...444000...000 with 2010 fours and 2010 zeros, if then you subtract m, you get 444...443666...666 like when you do 40-4=36

Answer 12

i got it .. thanks a lot guys ..