Iranian Mathematical Olympiad

No.

Let $p$ and $q$ be two distinct prime numbers. We define two sets as follows, $$\begin{array}{rl}A& =\{a_1,a_2,c_3,c_4,\dots ,c_{10}\},B=\{b_1,b_2,c_3,c_4,\dots ,c_{10}\}\\ a_1& =p{q}^{11},a_2=p^2{q}^{12},b_1=p^2{q}^{11},b_2=p{q}^{12},c_i=p^i{q}^i\end{array}$$ Now we create a one-to-one correspondence between numbers of superset of set $A$ and numbers of superset of set $B$ $$\begin{array}{rlrl}\gcd (c_i,c_j)& =\gcd (c_i,c_j),& \text{lcm}(c_i,c_j)& =\text{lcm}(c_i,c_j)\\ \gcd (a_1,a_2)& =p{q}^{11}=\gcd (b_1,b_2),& \text{lcm}(a_1,a_2)& =p^2{q}^{12}=\text{lcm}(b_1,b_2)\\ \gcd (a_1,c_i)& =p{q}^i=\gcd (b_1,c_i),& \text{lcm}(a_1,c_i)& =p^i{q}^{11}=\text{lcm}(b_2,c_i)\\ \gcd (a_2,c_i)& =p^2{q}^i=\gcd (b_2,c_i),& \text{lcm}(a_2,c_i)& =p^i{q}^{12}=\text{lcm}(b_1,c_i)\end{array}$$ Consequently, if these 90 numbers are written on the board, it is not possible to recognize the initial numbers for Masoud.