Observation: i) At the last step you will get the LCM of all the elements that is 2520. ii) You can't take the pairs (2,3) and (2,5) because condition of maximality will be violated. iii) At the first step possible GCD are 1,2,3. After first operation you will left with 9 elements. Calculation: i) After the first operation you have to choose the pairs cleverly such that after every operation the number of elements remains intact or number of elements is one less then the before. ii) After any arbitrary operations you have to stop at the 8th step. You will remain with 1,a,b,2520 where a,b are in divisor multiple relationship. Answer: So maximum number of operations Nahyun can perform is 8.

Rohit-2006 wrote: Observation: i) At the last step you will get the LCM of all the elements that is 2520. ii) You can't take the pairs (2,3) and (2,5) because condition of maximality will be violated. iii) At the first step possible GCD are 1,2,3. After first operation you will left with 9 elements. Calculation: i) After the first operation you have to choose the pairs cleverly such that after every operation the number of elements remains intact or number of elements is one less then the before. ii) After any arbitrary operations you have to stop at the 8th step. You will remain with 1,a,b,2520 where a,b are in divisor multiple relationship. Answer: So maximum number of operations Nahyun can perform is 8. I don't think so... At the test I found 17 possible

The final configuration should be as follows: \[ 7 \cdot 5 \cdot 3^2 \cdot 2^3, \; 5 \cdot 3 \cdot 2^2, \; 3 \cdot 2, \; 2, \; 2, \; 1, \; 1, \; 1, \; 1, \; 1 \] Now, let’s examine the operations involving numbers where one of them has \(7\) as a factor. Our strategy is to analyze the process by "removing" the factor \(7\) throughout all operations. Removing this factor reduces the possible operations slightly (for example, the operation on pairs \(7x\) and \(y\) where \(x | y\) would no longer apply). In each such an operation, the multiple of \(7\) gains at least one prime factor (counting multiplicities). Since the final form of the multiple of \(7\) is \(7 \cdot 5 \cdot 3^2 \cdot 2^3\), the number of such operations involving multiples of \(7\) does not exceed \(1 + 2 + 3 = 6\). Hence at most \(6\) such "additional" operations exist due to the presence of the factor \(7\). Next, consider operations on numbers after removing the factor \(7\). Now we examine the process by ignoring the factor \(5\). If both numbers are multiples of \(5\), the operation remains valid even if we "remove" the factor \(5\). We only need to consider cases involving pairs where only one number is a multiple of \(5\). The final forms of the multiples of \(5\) are \(5 \cdot 3^2 \cdot 2^3\) and \(5 \cdot 3 \cdot 2^2\), and initially, we had \(5 \cdot 2\), so the number of such operations does not exceed \(2 + 3 + 1 + 2 - 1 = 7\). Removing the factor \(5\) now leaves the numbers \(1, 1, 1, 2, 2, 4, 8, 3, 6, 9\). Note that \(7\) and \(10\) have been replaced by \(1\) and \(2\), respectively. For an operation on numbers of the form \(2^a 3^b\) and \(2^p 3^q\), the pairs \((a, b)\) and \((p, q)\) are arranged oppositely with respect to order. Thus, by the equality condition of rearrangement inequality, the sum of products of the exponents of \(2\) and \(3\) increases strictly with each operation. Initially, this sum is \(1\), and it increases to \(9\) in the final state. Therefore, the number of operations on these numbers does not exceed \(8\). Hence, the total number of operations does not exceed \(21\). Moreover, since for each of \(4\), \(8\), and \(9\), only one multiple of each exists at each step, at least three operations must apply to pairs involving multiples of both \(5\) and \(7\) simultaneously. Therefore, the total number of operations does not exceed \(21 - 3 = 18\). Finally, we construct an example sequence of operations to verify that achieving \(18\) operations is possible: \[ (9,6), \; (2,3), \; (4,6), \; (8,12), \; (2,3), \; (4,6), \; (12,18), \; (24,36), \; (6,10), \; (12,30), \; (60,72), \]\[ (7,2), \; (14,6), \; (12,42), \; (84,360), \; (2,5), \; (6,10), \; (12,30) \]This sequence shows that the maximum number of operations is indeed \(\boxed{18}\). - Solve with my collegue