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.
let y = 10^774then y^3 = 10^3*774 = 10^2322
So the expression can be written as ay^3 / (y + 3)
where a = 10^-100
Now expand 1/(y+3) in a taylor series:
1/(y+3) ~= 1/y - 3/y^2 + (term of order 1/y^3)
And so the expression itself is ~-
ay^2 - 3ay + (term of order a)
Given that a = 10^-100, this third term, whatever it is, is very much less than 1, and so disappears entirely inside the floor function
We're then left with 10^-100 * (10^1548 - 3*10^774)
= 10^1448 - 3*10^674
= 10^674 * (10^774 - 3)
That's a number that has 773 9's and a 7, and then 674 zeros tacked on, so the last two digits are both zero.
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Posted by Paul
on 2023-09-21 16:29:56 |
Analytic solution
