Final answer:
The greatest common divisor of the expressions 21n+4 and 14n+3 is 1 because the coefficients of n, after dividing both expressions by 7, are 3 and 2, which are relatively prime.
Step-by-step explanation:
To find the greatest common divisor (gcd) of the expressions 21n+4 and 14n+3, we can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the gcd of two numbers by repeatedly applying the division lemma. We start by performing polynomial division (if possible), but in this case we notice that both expressions share a common form: they can both be divided by 7, leaving us with 3n in the first expression and 2n in the second, plus the constants 4/7 and 3/7, respectively. Since constants don’t affect the gcd of variables, we focus on the coefficients of n, which are 3 and 2. Clearly, 3 and 2 are relatively prime, meaning they have no common divisor other than 1. Thus, the gcd of 21n+4 and 14n+3 is 1.