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.

Find all possible quadruplets (a,b,c,d) of positive integers, that satisfy this equation:

a!-b!-c! = d^{2}.

Prove that no further quadruplet satisfies the given conditions.

In the middle of 2023 i.e. between 20 and 23 enter
a number such that the new number will be a multiple of 2023.

a. What is the smallest number to fulfill the above requirement?

b. What if the number to be inserted should consist of the same digit repeated n times? Please provide a solution with lowest n.

c. In case no solution exists for (b) or it needs too much time, solve for 23, inserting a string of numbers between 2 and 3 so that the resulting number will be divisible by 23.

Find all possible (x,y) of 1 digit positive integers for which:

x

^{y}-y

^{x}=11y+x