As n ranges over the positive integers, what is the maximum possible value for the greatest common divisor of 11n + 3 and 2n + 1?

Guest Dec 29, 2021

#1**+1 **

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 p**ositive 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).

OrcSlop Dec 29, 2021