# 3 in 1

Algebra Level pending

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:

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

2. At an angle of $$120$$ degrees to the positive x-axis

3. 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:

1. $$f(m,n)=f(n,m)$$ for all $$n,m$$ in $$N$$

2. $$f(m,n)=f(m,m+n)$$ for all $$n,m$$ in $$N$$

3. $$f(n,0)=f(0,n)=n$$ 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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