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Determine the last two digits of this expression:

⌊(10^2222)/(10^774+3)⌋

Notes:
1) ⌊n⌋ 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! = d2.

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