As n ranges over the positive integers, what is the maximum possible value for the greatest common divisor of 11n + 3 and 2n + 1?
+20 5.
say X divides both 11n+3 and 2n+1, and is the greatest common divisor.
then we have the following system of equations
11n+3=ax
2n+1=bx
where a, b, and x are all positive integers.
then, multiplying 2n+1=bx by 3 and subtracting it from 11n+3=ax (elimination for a system of equations),
we have ax-3bx=5n
x(a-3b)=5n
n=x(a-3b)/5
using this value for n, we substitute it into the first equation 11n+3=ax
\(\frac{11x(a-3b)}{5}+3=ax \)
we simbplify this equation to get \(x(2a-11b)=5\)
However, because all x, a, and b are positive integers, we have either x=1, or x=5. The greatest value of x is 5.
we can note that when n=2, both 11n+3 and 2n+1 are divisible by 5 (25 and 5 respectively).
