Congratulations to Felix Reiter from Bottisham Village College for successfully solving this problem!
The key idea here is to split our problem into cases.
To solve the problem of finding two coprime integers 
and 
such that 
, we consider cases based on the properties of 
:
Case 1: is odd
Write 
for some integer 
.
Choose: 
, 
.
Since and are consecutive integers, they are coprime.
Case 2: is even and divisible by 
Write 
for some integer .
Choose: 
, 
.
These differ by 
, so their only possible common factor is , but both are odd, so they are coprime.
Case 3: is even but not divisible by
Write 
, where is odd.
Choose: 
, 
.
These differ by , so their only possible common factors are or , but is odd, so and are both odd and coprime.
