**Problem 1:**

There are 10 numbers \(\{2,3,5,7,11,13,17,19,23,29\} \) written on a blackboard. At each turn, any two numbers are chosen from the board and they are replaced by their G.C.D and L.C.M. If it is possible to get all odd numbers after a finite number of steps, the answer is 50, else the answer is 10. (Let the answer to this problem be \(Y\) )

**Problem 2:**

Let a person be at \(O(0,0)\) initially. He wants to go to his home at \(Q(1,0)\). He wants to reach his home in 10 steps, no more no less. (The segment \(OQ\) has length \(1\). The kind of steps he can take is a step of length \(1\) to any of the following three directions:

To the right, that is along the positive x-axis

At an angle of \(120\) degrees to the positive x-axis

At an angle of \(-120\) degrees to the positive x-axis

Let him take \(a\) steps of the first kind, \(b\) steps of the second kind, \(c\) steps of the third kind, and let the set \(S\) be the set containing \(a\), \(b\) and \(c\) only. (A set contains distinct elements only.)

Then the sum of the elements of \(S\) is \(X\).

**Problem 3:**

Let \(f:N^2->N\) where N is the set of non-negative integers be a function satisfying the following conditions:

\( f(m,n)=f(n,m)\) for all \(n,m\) in \(N\)

\(f(m,n)=f(m,m+n)\) for all \(n,m\) in \(N\)

\(f(n,0)=f(0,n)=n\) for all \(n\) in \(N\)

Find \(f((Y-1)^{\frac{3}{2}}+2, (X+1)^{\frac{2}{3}}.X+1)\). Let the answer be \(Z\).

Now the answer is \((Y^2-X^2)mod(Y).1000+X.100+Z\)

\(A \text{ (mod) } B\) refers to the remainder on dividing \(A\) by \(B\)

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