Determine the last two digits of this expression:
ā(10^2222)/(10^774+3)ā
Notes:
1) ānā is the floor of n, that is, the greatest integer less than or equal to n.
2) Computer program/excel solver assisted solutions are welcome, but a semi-analytic (p&p and hand calculator) methodology is preferred.
Rewrite 10^2222 as 10^674 * ((10^774)^2 - 9 + 9)
Substitute this form into the fraction:
10^674*((10^774)^2 - 9 + 9)/(10^774 + 3)
Now we can split this fraction into two parts:
10^674*((10^774)^2 - 9)/(10^774 + 3) + 9*10^674/(10^774 + 3)
The fraction on the left has a term that is a difference of squares: (10^774)^2 - 9 = (10^774 - 3)*(10^774 + 3)
Then that fraction reduces to an integer 10^674*(10^774 - 3)
Then note that 0 < 9*10^674/(10^774 + 3) < 1.
Then finally we can conclude that the floor of the original fraction is 10^674*(10^774 - 3).
This is obviously a multiple of 100, so the last two digits sought are 00.
Analytic Solution
