and 
then find the value of 
be the number of positive integers 
such that 
and 
divides
where 
denotes the number of positive integers that divide 
and 
.
.
Let 
.
consists of the union of disjoint pieces. Compute the distance between the endpoints of any one piece
Y by
Source: 2025 Nepal Mock TST Day 1 Problem 1
Prajit and Kritesh challenge each other with a marble game. In a bag, there are initially
red marbles and 
blue marbles. The rules of the game are as follows:
Move: In each turn, a player (either Prajit or Kritesh) removes two marbles from the bag.
If the two marbles are of the same color, they are both discarded and a red marble is added to the bag.
If the two marbles are of different colors, they are both discarded and a blue marble is added to the bag.
The game continues by repeating the above move.
Prove that no matter what sequence of moves is made, the process always terminates with exactly one marble left. In addition, find the possible colors of the marble remaining.
Prajit and Kritesh challenge each other with a marble game. In a bag, there are initially
red marbles and 
blue marbles. The rules of the game are as follows:Move: In each turn, a player (either Prajit or Kritesh) removes two marbles from the bag.
If the two marbles are of the same color, they are both discarded and a red marble is added to the bag.
If the two marbles are of different colors, they are both discarded and a blue marble is added to the bag.
The game continues by repeating the above move.
Prove that no matter what sequence of moves is made, the process always terminates with exactly one marble left. In addition, find the possible colors of the marble remaining.
at this point a player will remove the 2 remaining marbles and add one. The game then stops.
