Question

Ways of a number to be expressed as sum of coprime to each other

Answer 1

First suppose that 55 divides neither coprime factor. Then we have two cases-

Case 1: One factor must be 22 to some positive power while the other factor must be 33 to some positive power. Hence there are 3⋅2=63⋅2=6 of these (as there are 33 possible positive powers of 22 and there are 22 possible positive powers of 33).

Case 2: One factor is 11 while the other factor is a divisor of 23322332. Hence there are (3+1)(2+1)=12(3+1)(2+1)=12 of these.

Now, note that all of these pairs consist of different numbers except the pair {1,1}{1,1}. For each of the pairs with two distinct elements, we can multiply either element by 55 to get two new coprime pairs of factors (that is the solution {a,b}{a,b} gives rise to the solutions {5a,b}{5a,b} and {a,5b}{a,5b}). Finally, the pair {1,1}{1,1} under this process only gives us one new pair: {1,5}{1,5}.

In conclusion, we have

3(6+12−1)+2=53

3(6+12−1)+2=53

coprime pairs.