As n ranges over the positive integers, what is the maximum possible value that the greatest common divisor of 13n+8 and 5n+3 can take?
The maximum value of the GCD ==1. Because the two expressions are always "relatively prime" to each other, no matter what the value of n is. Therefore, their GCD will always be== 1.
Suppose that 13n + 8 and 5n + 3 have a factor p in common.
Let 13n + 8 = sp and 5n + 3 = tp for some numbers s and t.
Eliminate n between the two equations, (5 times the first minus 13 times the second) and you arrive at
1 = 5sp - 13tp = p(5s - 13t).
Since we are talking integers, it follows that p = 1.